covariant

LL-Q1860 (eng)-Flame, not lame-covariant.wav

(algebra) A bihomogeneous polynomial in x, y, ... and the coefficients of some homogeneous form in x, y, ... that is invariant under some group of linear transformations.

(algebra) The variety defined by a covariant.


contravariant

LL-Q1860 (eng)-Flame, not lame-contravariant.wav

(algebra) A bihomogeneous polynomial in dual variables of x, y, ... and the coefficients of some homogeneous form in x, y, ... that is invariant under some group of linear transformations.


spinor

LL-Q1860 (eng)-Flame, not lame-spinor.wav

(algebra) An element of the fundamental representation of a Clifford algebra that transforms to its negative when the space is rotated through a complete turn from 0° to 360°.


coset

LL-Q1860 (eng)-Flame, not lame-coset.wav

From co- + set; apparently first used 1910 by American mathematician George Abram Miller.

1970 [Addison Wesley], Frederick W. Byron, Robert W. Fuller, Mathematics of Classical and Quantum Physics, Volumes 1-2, Dover, 1992, page 597,
Theorem 10.5. The collection consisting of an invariant subgroup H and all its distinct cosets is itself a group, called the factor group of G, usually denoted by G/H. (Remember that the left and right cosets of an invariant subgroup are identical.) Multiplication of two cosets aH and bH is defined as the set of all distinct products z = xy, with x ∈ aH and y ∈ bH; the identity element of the factor group is the subgroup H itself.

1982 [Stanley Thornes], Linda Bostock, Suzanne Chandler, C. Rourke, Further Pure Mathematics, Nelson Thornes, 2002 Reprint, page 614,
In general, the coset in row x consists of all the elements xh as h runs through the various elements of H.

Example 3. Let G#61;#92;mathbb#123;Z#125; (the operation is #43;), H#61;2#92;mathbb#123;Z#125;. Then the coset 1#43;2#92;mathbb#123;Z#125; is the set of integers of the form 1#43;2k where k runs through all elements of #92;mathbb#123;Z#125;.

2009, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 3rd edition, Springer, page 231:

(algebra, group theory) The set that results from applying a group's binary operation with a given fixed element of the group on each element of a given subgroup.

kõrvalklass

result of applying group operation with fixed element from the parent group on each element of a subgroup


left coset

(algebra) A left coset of a subgroup is a copy of that subgroup, multiplied on the left by some element from the parent group


right coset

(algebra) A copy of a subgroup, multiplied on the right by some element from the parent group.


multiplicative

LL-Q1860 (eng)-Flame, not lame-multiplicative.wav

From 6th-century Latin multiplicativus, from Classical Latin multiplicāre (“to multiply”).

(algebra) Having multiplication as an operator.


wedge product

(algebra) A way of representing an exterior product using a wedge symbol: ∧.


dyadic operation

en-noun

dyadic operation (plural dyadic operations)

(algebra) An operation taking two operands.


unary operation

unary operation (plural unary operations)

(algebra) An operation taking one operand, such as the square, factorial or absolute value operation.


unary operator

(algebra, computing) A symbol representing unary operation; an operator taking one operand.


binary operator

binary operator (plural binary operators)

The "four rules" +, -, × and ÷ are examples of binary operators.

(algebra, computing) A symbol representing binary operation; an operator taking two operands.


exact sequence

(algebra) A sequence of modules (or groups, vector spaces, etc.; most generally, objects of an abelian category) with adjacent objects connected by homomorphisms (generally, morphisms) such that the image of one homomorphism is the kernel of the next.


trinomial

LL-Q1860 (eng)-Flame, not lame-trinomial.wav

(algebra) An expression consisting of three terms.


associativity

LL-Q1860 (eng)-Flame, not lame-associativity.wav

(algebra) The condition of being associative.

結合法則 (ketsugō hōsoku, literally “associative law”)


abelian group

abelian groups (plural)
Abelian group (alternative)

Named in honour of Niels Henrik Abel (1802–1829), a Norwegian mathematician.

1986, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, 2010 softcover reprint, page 12,
Let G and H be partially ordered abelian groups. A positive homomorphism from G to H is any abelian group homomorphism f:G→H that maps positive elements to positive elements, that is, f(G⁺)⊆H⁺.

2000, David Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, Softcover reprint of 1st edition, page 74,
Chapter 2 is a brief introduction to some fundamental techniques for countable torsion-free abelian groups.

By the end of Chapter 2 we had the full power of the Pontryagin Duality Theorem for compact abelian groups and for discrete abelian groups. Locally compact abelian groups are much closer to compact abelian groups than is apparent at first sight.

2013, Karl H. Hofmann, Sidney A. Morris, The Structure of Compact Groups: A Primer for the Student: A Handbook for the Expert, Walter de Gruyter, page 299:

(algebra) A group in which the group operation is commutative.


nilpotent

LL-Q1860 (eng)-Flame, not lame-nilpotent.wav

From nil (“not any”) + potent (“having power”) with literal meaning “having zero power” - bearing Latin roots nil and potens.
Coined in 1870, along with idempotent, by American mathematician Benjamin Peirce to describe elements of associative algebras.

compound

If a square matrix is upper triangular, then it is nilpotent (under the usual matrix multiplication).

The rest of this book is devoted to determining the conjugacy classes and centralizers of nilpotent elements in L(G) and unipotent elements in G, where G is an exceptional algebraic group of type E₈,E₇, E₆, F₄ or G₂ over an algebraically closed field K of characteristic p. This chapter contains statements of the main results for nilpotent elements.

2012, Martin W. Liebeck, Gary M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, American Mathematical Society, page 129:

(mathematics, algebra, ring theory, of an element x of a ring) Such that, for some positive integer n, xⁿ = 0.

linear transformation
linear transformation

(mathematics, algebra) In any of several technical senses: behaving analogously to nilpotent ring elements as an element of some other algebraic structure; composed of elements displaying such behavior.

(Lie theory, of an element x of a Lie algebra L) Belonging to the derived algebra of L and such that the adjoint action of x is nilpotent (as a linear transformation on L).

lower central series
lower central series

(Lie theory, of a Lie algebra) Such that the lower central series terminates.

(group theory, of a group) Admitting a central series of finite length.

(ring theory, of an ideal I) Such that there exists a natural number k with Iᵏ = 0.

(semigroup theory, of a semigroup with zero) Containing only nilpotent elements.

(of an algebra over a commutative ring) Such that there exists some natural number n (called the index of the algebra) such that all products (of elements in the given algebra) of length n are zero.


nilpotent

The so-called spinor algebra of C(2), the language of the quantum mechanics, is formulated in terms of the idempotents and nilpotents of the geometric algebra of space, including its beautiful representation on the Riemann sphere, and a new proof of the Heisenberg uncertainty principle.

2015, Garret Sobczyk, “Part I: Vector Analysis of Spinors”, in arXiv:


number field

algebraic number field
algebraic number field

(algebra, field theory) algebraic number field


division ring

multiplicative inverse
multiplicative inverse

(algebra) A ring with 0 ≠ 1, such that every non-zero element a has a multiplicative inverse, meaning an element x with ax = 1.


coprime

LL-Q1860 (eng)-Flame, not lame-coprime.wav

greatest common divisor
greatest common divisor

(algebra, by extension, of two or more polynomials) Whose greatest common divisor is a nonzero constant (i.e., polynomial of degree 0).


Cayley table

multiplication table
multiplication table

(algebra) The analogue for a group of a multiplication table.


quadrinomial

LL-Q1860 (eng)-Flame, not lame-quadrinomial.wav

(algebra) An expression consisting of four terms.


trichotomy

en-noun

trichotomy (countable and uncountable, plural trichotomies)

LL-Q1860 (eng)-Flame, not lame-trichotomy.wav

From Ancient Greek τριχοτομία (trikhotomía), from τρίχα (tríkha) + τόμος (tómos) + -ία (-ía).

affix

[A linear order] obeys the following trichotomy law[.]

Aliprantis, Border (2006), Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.), p. 6

(algebra) the property of an order relation whereby, given an ordered pair of elements (of a given algebraic structure), exactly one of these is true: the first element is 'less than' the second one, the second is 'less than' the first, or the two elements are equal.


algebraic structure

en-noun

algebraic structure (plural algebraic structures)

(algebra, universal algebra) Any one of the numerous types of mathematical object studied in algebra and especially in universal algebra.


homomorphic

en-adj

homomorphic (comparative more homomorphic, superlative most homomorphic)

more homomorphic (comparative)
most homomorphic (superlative)

LL-Q1860 (eng)-Flame, not lame-homomorphic.wav

If it is necessary or desirable to preserve input or output port structure in homomorphic images, one must insist on a special case of homomorphism.

1993, A. Wayne Wymore, Model-Based Systems Engineering, CRC Press, →ISBN, page 182:

In this chapter two types of signature schemes satisfying homomorphic properties are presented. In the first section a description of the homomorphic signature schemes suitable in the single-user scenario is provided.

2016, Giulia Traverso, Denise Demirel, Johannes Buchmann, Homomorphic Signature Schemes: A Survey, Springer, →ISBN, page 11:

Fully homomorphic encryption (FHE) is a new encryption technique that enables computing on data blindly, without decrypting it.

2023 May 26, Rand Hindi, “How Homomorphic Encryption Can Make Blockchain Private”, in Forbes, New York, N.Y.: Forbes Media, →ISSN, →OCLC, archived from the original on 2023-06-12:

(algebra, biology) Of or pertaining to homomorphism; having a homomorphism.


rng

From ring, with i dropped to suggest the absence of the identity element.

(algebra) An algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element.


semiring

LL-Q1860 (eng)-Flame, not lame-semiring.wav

(algebra) An algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.


complete lattice

en-noun

complete lattice (plural complete lattices)

partially ordered set
partially ordered set

(algebra) A partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).


algebraically closed

The fundamental theorem of algebra states that the field of complex numbers, #92;mathbb#123;C#125;, is algebraically closed.

Definition. A field is algebraically closed when it satisfies the equivalent conditions in Proposition 4.1. #92;square
For instance, the fundamental theorem of algebra (Theorem III.8.11) states that #92;Complex is algebraically closed. The fields #92;R, #92;Q, #92;Z#95;p, are not algebraically closed, but #92;R and #92;Q can be embedded into the algebraically closed field #92;Complex.

2007, Pierre Antoine Grillet, Abstract Algebra, 2nd edition, Springer, page 166:

In many ways #92;Q#95;p is analogous to #92;R. For example, #92;R is not algebraically closed. The exercises below show that #92;Q#95;p is not algebraically closed. However, by adjoining i#61;#92;sqrt#123;-1#125; to #92;R, we get the field of complex numbers, which is algebraically closed. In contrast, the algebraic closure #92;overline#92;Q#95;p of #92;Q#95;p is not of finite degree over #92;Q. Moreover, #92;Complex is complete with respect to the extension of the usual norm of #92;R. Unfortunately, #92;overline#92;Q#95;p is not complete with respect to the extension of the p-adic norm. So after completing it (via the usual method of Cauchy sequences) we get a still larger field, usually denoted by #92;Complex#95;p, and it turns out to be both algebraically closed and complete.

2008, M. Ram Murty, Problems in Analytic Number Theory, 2nd edition, Springer, page 155:

Definition 5.3.14. Let F be a field. A field extension P is called an algebraic closure of F if P is algebraically closed and every proper subfield of P containing F is not algebraically closed.
In other words, the algebraic closure of F is the minimal algebraically closed field containing F.

2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 195:

(algebra, field theory, of a field) Which contains as an element every root of every nonconstant univariate polynomial definable over it (i.e., over said field).

algébriquement clos

that contains among its elements every root of every nonconstant single-variable polynomial definable over it

(algebra, group theory, of a group) Such that any finite set of equations and inequations has a solution.


quotient space

Thus if #92;mathfrak#123;D#125; is an arbitrary decomposition of a space X into mutually disjoint subsets, then the corresponding quotient space is obtained by 'shrinking' or 'identifying' each member of #92;mathfrak#123;D#125; to a single point. For this reason, the quotient spaces are sometimes called identification spaces and quotient maps as^([sic]) identification maps.

1983, K. D. Joshi, Introduction to General Topology, New Age International, page 129:

1989, unnamed translator, Nicolas Bourbaki, Elements of Mathematics: General Topology: Chapters 1-4, [1971, N. Bourbaki, Éléments de Mathématique: Topologie Générale 1-4, Masson], Springer, page 110,
PROPOSITION 6. Every quotient space of a connected space is connected.

The quotient space obtained from such a relation is called extended Reeb (ER) quotient space. Moreover, the ER quotient space, which is an abstract sub-space of M* and is independent from the geometry, may be represented as a traditional graph, which is called the extended Reeb graph (ERG).

2004, Silvia Biasotti, Bianca Falcidieno, Michela Spagnuolo, “6: Surface Shape Understanding Based on Extended Reeb Graphs”, in Sanjay Rana, editor, Topological Data Structures for Surfaces: An Introduction to Geographical Information Science, Wiley, page 96:

equivalence relation
equivalence relation

(topology and algebra) A space obtained from another by identification of points that are equivalent to one another in some equivalence relation.

identification space

space composed of points equivalent to each other in some relation


groupoid

LL-Q1860 (eng)-Flame, not lame-groupoid.wav

(algebra) A magma: a set with a total binary operation.

[...] A category in which every morphism is an isomorphism is called a groupoid.
 This means that a group is a groupoid with only one object. [...]

(algebra and category theory) A set with a partial binary operation that is associative and has identities and inverses.

groupoïde

a set with a partial binary operation...


nonassociative

LL-Q1860 (eng)-Flame, not lame-nonassociative.wav

This book has been written about associative algebras. However, there is an analogous “PI-theory” that can be carried out for nonassociative rings, which is outlined here.

1980, Lewis Halle Rowen, Polynomial Identities in Ring Theory, Academic Press, page 327:

1996, E. G. Goodaire, E. Jespers, C. Polcino Milies, Alternative Loop Rings, Elsevier, page 5,
Two important functions in nonassociative ring theory are the commutator and associator which, for elements a, b, c in a ring are defined respectively by
[a, b] = ab − ba
and
[a, b, c] = a(bc) − (ab)c.

2012, W. B. Vasantha Kandasamy, Florentin Smarandache, Non Associative Algebraic Structures Using Finite Complex Numbers, Zip Publishing, page 5,
Authors in this book for the first time have constructed nonassociative structures like groupoids, quasi loops, non associative semirings and rings using finite complex modulo integers.

(algebra, of an algebraic structure) Whose multiplication operation is not assumed to be associative for all elements.


normalizer

LL-Q1860 (eng)-Flame, not lame-normalizer.wav

(algebra) The subset of elements of some group which leave some given subset invariant when conjugating it.

Normalisator

set of group elements leaving a given subset invariant under conjugation


commutant

LL-Q1860 (eng)-Flame, not lame-commutant.wav

The basic mathematical entity to be used here in elucidating the different senses of superselection rules is a von Neumann algebra #123;#92;mathfrak#123;M

2008 September 27, John Earman, “Superselection Rules for Philosophers”, in Erkenntnis, volume 69, number 3, →DOI:

(algebra, logic) The subset of all elements of a semigroup that commute with the elements of a given subset


locally indicable

(algebra, of a group) Such that each of its nontrivial, finitely generated subgroups maps homomorphically onto ℤ.


alternative algebra

alternative algebra (plural alternative algebras)

(algebra) An algebra such that every subalgebra generated by two elements is associative.


power-associative algebra

power-associative algebra (plural power-associative algebras)

(algebra) An algebra such that every subalgebra generated by one element is associative.


Boolean lattice

Boolean lattice (plural Boolean lattices)

A Boolean lattice always has 2ⁿ elements for some cardinal number n.

If two Boolean lattices have the same size, then they are isomorphic.

(algebra) The lattice corresponding to a Boolean algebra.


distributive lattice

distributive lattice (plural distributive lattices)

In a distributive lattice, the set of complemented elements is a sublattice.

(algebra) A lattice in which the "meet" and the "join" operations are distributive with respect to each other.


adjunction

adjunction (countable and uncountable, plural adjunctions)

LL-Q1860 (eng)-Flame, not lame-adjunction.wav

From Latin adjunctio, from adjungere: compare French adjonction, and see adjunct.

The ring obtained after the adjunction of the elements a,b and y to the ring R may be denoted R#91;a,b,y#93;.

The field adjunction #92;mathbb#123;Q#125;(#92;pi) can be obtained from #92;mathbb#123;Q#125; by adjoining #92;pi to #92;mathbb#123;Q#125;.

(mathematics, chiefly algebra and number theory) The process of adjoining elements to an algebraic structure (usually a ring or field); the result of such a process.


additive group

additiivinen ryhmä

group or algebraic object regarded as a group with operation called addition

(mathematics) Any group, or algebraic object that can be regarded as a group, whose binary operation is called addition and denoted +; specifically:

(algebra) Any of certain algebraic objects (ring, field or vector space, etc., whose definition includes a commutative operation called addition) regarded as a group under addition.


derived group

(algebra) The subgroup of a specified group generated by the larger group's commutators.


commutator subgroup

commutator subgroup (plural commutator subgroups)

The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian.^([wikipedia])


group ring

(algebra) Given ring R with identity not equal to zero, and group G=g_1,g_2,...,g_n, the group ring RG has elements of the form a_1g_1+a_2g_2+...+a_ng_n (where a_i isin R) such that the sum of a_1g_1+a_2g_2+...+a_ng_n and b_1g_1+b_2g_2+...+b_ng_n is (a_1+b_1)g_1+(a_2+b_2)g_2+...+(a_n+b_n)g_n and the product is ∑ₖ₌₁ⁿ(∑_(g_ig_j=g_k)a_ib_j)g_k.


integral domain

integral domain (plural integral domains)

A ring R is an integral domain if and only if the polynomial ring R#91;x#93; is an integral domain.

For any integral domain there can be derived an associated field of fractions.

For integral domains, we will use a⁻¹ to designate the multiplicative inverse of a (if it has one; since not all elements need have inverses, this notation can be used only where it can be shown that an inverse exists).

1990, Barbara H. Partee, Alice ter Meulen, Robert E. Wall, Mathematical Methods in Linguistics, Kluwer Academic Publishers, page 266:

An integral domain is said to have strong pseudo-Dedekind factorization if each proper ideal can be factored as the product of an invertible ideal (possibly equal to the ring) and a finite product of pairwise comaximal prime ideals with at least one prime in the product.

2013, Marco Fontana, Evan Houston, Thomas Lucas, Factoring Ideals in Integral Domains, Springer, page 95:

#91;#92;mathbb#123;Z#125;#59;#43;,#92;cdot#93;, #91;#92;mathbb#123;Z#125;#95;p,#43;#95;p,#92;times#95;p#93; with p a prime, #91;#92;mathbb#123;Q#125;#59;#43;,#92;cdot#93;, #91;#92;mathbb#123;R#125;#59;#43;,#92;cdot#93;, and #91;#92;mathbb#123;C#125;#59;#43;,#92;cdot#93; are all integral domains. The key example of an infinite integral domain is #91;#92;mathbb#123;Z#125;#59;#43;,#92;cdot#93;. In fact, it is from #92;mathbb#123;Z#125; that the term integral domain is derived. Our main example of a finite integral domain is #91;#92;mathbb#123;Z#125;#95;p,#43;#95;p,#92;times#95;p#93;, when p is prime.

2017, Ken Levasseur, Al Doerr, Applied Discrete Structures: Part 2 - Applied Algebra, Lulu.com, page 171:

(algebra, ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero.


Gaussian integer

Gaussian integer (plural Gaussian integers)

We could say that a Gaussian integer is larger than another if its norm is larger, that is, if its distance from the origin is larger. That's all right, although it has the peculiarity that 5 is larger than -7 as an integer, but smaller than -7 as a Gaussian integer! In a way, this notion of larger makes more sense: shouldn't -7 be considered larger than 5?

1997, Bernard L. Johnston, Fred Richman, Numbers and Symmetry: An Introduction to Algebra, CRC Press, page 44:

2000, André Weilert, Asymptotically fast GCD Computation in ℤ[i], Wieb Bosma (editor), Algorithmic Number Theory: 4th International Symposium, ANTS-IV, Proceedings, Springer, LNCS 1838, page 595,
We present an asymptotically fast algorithm for the computation of the greatest common divisor (GCD) of two Gaussian integers.

2008, Timothy Gowers, June Barrow-Green, Imre Leader (editors), The Princeton Companion to Mathematics, Princeton University Press, page 319,
For example, in the ring of Gaussian integers, R_-1, we have the factorizations
2=(1+i)(1-i),
5=(1+2i)(1-2i),
13=(2+3i)(2-3i),
17=(1+4i)(1-4i),
29=(2+5i)(2-5i),
⋮
where all the Gaussian integer factors in brackets above are irreducible elements of the ring of Gaussian integers.

(algebra) Any complex number of the form a + bi, where a and b are integers.


nonpolynomial

LL-Q1860 (eng)-Flame, not lame-nonpolynomial.wav


nonpolynomial

(algebra) An expression that is not a polynomial.


left ideal

If I is a left ideal of R then rI⊆I for any r isin R.

(algebra) A subset of a ring which is closed under left-multiplication by any element of the ring.


right ideal

If I is a right ideal of R then Ir⊆I for any r isin R.

(algebra) A subset of a ring which is closed under right-multiplication by any element of the ring.


two-sided ideal

two-sided ideal (plural two-sided ideals)

(algebra) A subset of a ring which is both a left ideal and a right ideal. Commonly referred to simply as an ideal, but referred to as two-sided ideal for emphasis.


principal ideal

principal ideal (plural principal ideals)

(algebra) An ideal I in an algebraic object R (which could be a ring, algebra, semigroup or lattice) that is generated by a given single element a ∈ R; the smallest ideal that contains a.

two-sided principal ideal

smallest ideal that contains a given element


principal ideal domain

principal ideal domain (plural principal ideal domains)

(algebra) An integral domain in which every ideal is a principal ideal.


semisimple

LL-Q1860 (eng)-Flame, not lame-semisimple.wav

(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.

(category theory, most generally, of an abelian category) Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects.

(module theory, of a module) In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules.

(ring theory, of an algebra or ring) Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.

(of a ring, somewhat proscribed) Semiprimitive: having trivial Jacobson radical.

(linear algebra, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.

(Lie theory, of a Lie algebra) Being a direct sum of simple Lie algebras.

linear representation
linear representation

simple representation
simple representation

irreducible representation
irreducible representation

(representation theory, of a linear representation of a group or algebra) Being a direct sum of simple representations (also known as irreducible representations).

(group theory, of an algebraic group) Being a linear algebraic group whose radical of the identity component is trivial.


associator

(algebra) A multilinear map, given by [x, y, z] = xy(z) − x(yz), that measures the degree of nonassociativity of a ring or algebra.

Assoziator

multilinear map measuring the degree of nonassociativity


PID

principal ideal domain
principal ideal domain#English

(algebra) Initialism of principal ideal domain.


divisibility sequence

divisibility sequence (plural divisibility sequences)

If #92;mathcalC#61;(C#95;n)#95;#123;n#92;ge 1#125; is any divisibility sequence, one says that a prime p is a primitive divisor of C#95;n if p#92;vertC#95;n but p#92;nmidC#95;1C#95;2C#95;#123;n-1#125;. Primitive divisors of certain divisibility sequences were studied by Zsigmondy [37] in the 19ᵗʰ century.

2012, P. Ingram, J. H. Silverman, “Primitive Divisors in Elliptic Divisibility Sequences”, in Dorian Goldfeld, Jay Jorgenson, Peter Jones, Dinakar Ramakrishnan, Kenneth Ribet, John Tate, editors, Number Theory, Analysis and Geometry, Springer,, page 244:

These divisibility sequences satisfy the same recurrence relations as the polynomials #92;phi#95;n and #92;psi²#95;n (see Appendix C).

2013, Graham Everest, Thomas Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, page 138:

Moreover, in order to keep a divisibility sequence normalized, we can assume without loss of generality that a#95;0 = 0 and a#95;1#61;1.

2021, Masum Billal, Samin Riasat, Integer Sequences, Springer, page 59:

(number theory, algebra) Any sequence of integers {aₙ}, indexed by the natural numbers, such that if n is divisible by m then aₙ is divisible by aₘ.


collineation

collineation (countable and uncountable, plural collineations)

LL-Q1860 (eng)-Flame, not lame-collineation.wav

From Latin collineare to direct in a straight line. See collimation.

(algebra) An isomorphism from one projective space to another, often to itself.


connex

LL-Q1860 (eng)-Flame, not lame-connex.wav

(algebra) A correspondence between a projective space and its dual.


ultraradical

ultraradicals (plural)
ultra-radical (alternative)

LL-Q1860 (eng)-Flame, not lame-ultraradical.wav

(algebra) A root of the polynomial x⁵ + x + a, where a is a complex number.


antidivision

antidivision (usually uncountable, plural antidivisions)

LL-Q1860 (eng)-Flame, not lame-antidivision.wav

2008, Patrick Bosc, Olivier Pivert, A Family of Tolerant Antidivision Operators for Database Fuzzy Querying Book Series:

It can be noticed that an antidivision is nothing but an antijoin followed by a projection over X […]

2008, Sergio Greco, Thomas Lukasiewicz, Scalable uncertainty management:

(algebra) A particular extension of the antijoin operator.


antijoin

LL-Q1860 (eng)-Flame, not lame-antijoin.wav

(algebra, databases) A type of join, denoted by ▷, such that R▷S yields only those tuples in R for which S has no corresponding tuple of the same value.


ordered field

(algebra) A field which has an order relation satisfying these properties: trichotomy, transitivity, preservation of an inequality when the same element is added to both sides, and preservation of an inequality when the same strictly positive element is multiplied to both sides.

corpo ordenado

field satisfying the properties of trichotomy, transitivity and preservation of an inequality when the same element is added to both sides


idealizer

LL-Q1860 (eng)-Flame, not lame-idealizer.wav

(algebra) For a ring R and an ideal m of R, the set of all u ∈ Frac(R) such that um ⊂ m, where Frac(R) is the fraction field of R.


scalar multiplication

scalar multiplication (countable and uncountable, plural scalar multiplications)

(abstract algebra) multiplication of a module element by a ring element


compositum

LL-Q1860 (eng)-Flame, not lame-compositum.wav

1998, Iain T. Adamson (translator), David Hilbert, The Theory of Algebraic Number Fields, [1897, D. Hilbert, Zahlbericht], Springer, page 98,
Of particular interest is the case in which the discriminants of the fields forming the compositum are relatively prime.

For #92;Q, we get the maximal abelian extension by adjoining all roots of unity, i.e., taking compositum of all m-th cyclotomic fields (this is the famous Kronecker-Weber theorem).

2004, Dinesh S. Thakur, Function Field Arithmetic, World Scientific, page 81:

This is then just the field compositum of all the quadratic extensions F(#92;sqrt#123;a#95;i#125;) in the algebraic closure of F.

2005, T. Y. Lam, Introduction to Quadratic Forms over Fields, American Mathematical Society, page 333:

(algebra, field theory) Given a field extension F/K and subextensions A and B, the smallest subextension that contains both A and B.

field compositum

smallest subextension of a given field extension that contains two given subextensions


Boolean ring

Named after George Boole (1815–1864), an English mathematician, educator, philosopher and logician.

From the defining idempotency property of a Boolean ring it is possible to prove that such ring has to have the further properties that each element is its own inverse and that such ring must be commutative.

multiplicative operation
multiplicative operation

(algebra) A ring whose multiplicative operation is idempotent.


multibracket

LL-Q1860 (eng)-Flame, not lame-multibracket.wav

(algebra) A partition of a set into disjoint subsets such that each subset contains an even number of elements, and that if those elements are ordered, they divide the subset into ranges (1st to 2nd element, 3rd to 4th element, and so on) such that no element from another subset appears in any of the ranges.


commutative algebra

commutative algebra (countable and uncountable, plural commutative algebras)

noncommutative algebra

concerned with rings that are not assumed to be commutative

Theorem. If the complex of the linearly independent numbers of the form #92;zetaa-a#92;zeta be deleted from an algebra, the remaining numbers form a commutative algebra.

1907, James Byrnie Shaw, Synopsis of Linear Associative Algebra, Carnegie Institution of Washington, page 46:

The existence of commutative algebras of Toeplitz operators is not usual, for example there are not non trivial commutative algebras of Toeplitz operators acting on the Hardy space.

2010, M. Loaiza, A. Sánchez-Numgaray, “On C*-Algebras of Super Toeplitz Operators with Radial Symbols”, in Roland V. Duduchava, Israel Gohberg, Sergei M. Grudsky, Vladimir Rabinovich, editors, Recent Trends in Toeplitz and Pseudodifferential Operators, Springer, page 175:

We devote this chapter to the study of unitary commutative algebras in dg-modules, in simplicial modules and in cosimplicial modules.

2017, Benoit Fresse, Homotopy of Operads and Grothendieck-Teichmuller Groups: Part 2, American Mathematical Society, page 163:

(algebra) Any algebra (mathematical structure) in which the multiplication operation is commutative.

algebra commutativa

algebra in which multiplication is commutative

Abelian algebra

algebra which has a commutative multiplication operation


Heyting algebra

Heyting algebra (plural Heyting algebras)

After Dutch mathematician Arend Heyting, who developed the theory as a way of modelling his intuitionistic logic.

The laws of Heyting algebra embody a rich and profound mathematical structure that is manifest in a variety of contexts. It arises from the epistemological deliberations of Brouwer, the topologisation (localisation) of set-theoretic notions, and the categorial formulation of set theory, all of which, although interrelated, are independently motivated. The ubiquity lends weight, not to the suggestion that the correct logic is in fact intuitionistic instead of classical, but rather to the recognition that thinking in such terms is simply inappropriate — in the same way that it is inappropriate to speak without qualification about the correct geometry.

1984, Robert Goldblatt, Topoi, the categorial analysis of logic, page xii:

which is just obvious.

1994, Francis Borceux, Handbook of Categorical Algebra 3: Categories of Sheaves, Cambridge University Press, page 13, Proposition 1.2.14 should certainly be completed by the observation that the modus ponens holds as well in every Heyting algebra. Since, in the intuitionistic propositional calculus, being a true formula is being a terminal object (see proof of 1.1.3), the modus ponens of a Heyting algebra reduces to a=1 and a<b imply b=1

1997, J. G. Stell, M. W. Worboys, The Algebraic Structure of Sets and Regions, Stephen C. Hirtle, Andrew U. Frank (editors), Spatial Information Theory A Theoretical Basis for GIS: International Conference, Proceedings, Springer, LNCS 1329, page 163,
The main contention of this paper is that Heyting algebras, and related structures, provide elegant and natural theories of parthood and boundary which have close connections to the above three ontologies.

(algebra, order theory) A bounded lattice, L, modified to serve as a model for a logical calculus by being equipped with a binary operation called "implies", denoted → (sometimes ⊃ or ⇒), defined such that (a→b)∧a ≤ b and, moreover, that x = a→b is the greatest element such that x∧a ≤ b (in the sense that if c∧a ≤ b then c ≤ a→b).

Arend Heyting
Heyting algebra
intuitionistic logic

algebra di Heyting

bounded lattice equipped with operation called implies


orthotomic

LL-Q1860 (eng)-Flame, not lame-orthotomic.wav

(algebra) imaginary; involving the square root of a negative number.


osculant

LL-Q1860 (eng)-Flame, not lame-osculant.wav

From Latin osculans, osculantis, present participle of osculari (“to kiss”). See osculate.

(algebra) The condition that the solution to a set of simultaneous quantics, is also the solution of the corresponding set of tangential quantics.


p-adic norm

Let K be a field with discrete valuation v, and #92;mathcal#123;O#125;#95;K be the valuation ring.
Definition 2.3.1 A p'''-adic norm on vector space V over K is a function #92;alpha#58;V#92;rightarrow#92;R satisfying:
a) #92;alpha(x)#92;ge 0 and #92;alpha(x)#61;0 if and only if x#61;0.
b) #92;alpha(ax)#61;#92;verta#92;vert#92;alpha(x) for a#92;inK and x#92;inV.
c) #92;alpha(x#43;y)#92;le#92;operatorname#123;sup#125;(#92;alpha(x),#92;alpha(y)) for x,y#92;inV.
If #92;alpha is a p'''-adic norm and tgt;0, then the dilation t#92;alpha is a p'''-adic norm, and we denote by N#95;p(V) the set of dilation classes of p'''-adic norms on V.

2006, Kang Zuo, Representations of Fundamental Groups of Algebraic Varieties, Springer, page 20:

(algebra) A norm on a vector space which is defined over a field equipped with a discrete valuation (a generalisation of p-adic absolute value).


nomial

LL-Q1860 (eng)-Flame, not lame-nomial.wav


homomorphous

en-adj

homomorphous (comparative more homomorphous, superlative most homomorphous)

more homomorphous (comparative)
most homomorphous (superlative)

LL-Q1860 (eng)-Flame, not lame-homomorphous.wav

From Ancient Greek ὁμός (homós, “same”) + μορφή (morphḗ, “shape”) + -ous.

(algebra, biology) Characterized by homomorphism.


inverse system

contravariant functor
contravariant functor

(algebra) A set of algebraic structures (which are part of a concrete category; e.g., groups) and a set of morphisms between them (e.g., group homomorphisms) which all together form a small category which is the image of a contravariant functor whose domain is a directed poset.


inverse limit

An inverse limit has “natural projections” which are restrictions of the projections of the Cartesian product (to a domain which is the inverse limit). The reason why the projections are described as “natural” would be the following: besides the functor from an index poset to the inverse system, there is another functor from the same index poset to the inverse limit of that system, this functor being a constant functor. Then there is a natural transformation from the constant functor to the inverse limit’s functor: the components of such natural transformation are the said “natural projections”.

Inverse limits are concrete-categorical versions of limits.

(algebra) A subset of the Cartesian product of all the members of an inverse system, such that a member M of the subset is a sort of “cross section” of the inverse system (as fiber bundle) induced by the morphisms of it. (If i<j in the indexing poset then f_ij:A_j→A_i in the inverse system and if a_i∈A_i, a_j∈A_j are components of M then f_ij(a_j)=a_i).


extension field

en-noun

extension field (plural extension fields)

Suppose F is a subfield of the field K. Then K is called an extension field of F. So, for instance, GF(4) and GF(8) are extension fields of GF(2), although GF(8) is not an extension field of GF(4).

1992, James G. Oxley, “Matroid Theory”, in Paperback, Oxford University Press, published 2006, page 215:

This extension field of F always contains a root of f in the sense that if K#61;F#91;x#93;#47;(f(x)) then x is a root of f(y) in K#91;y#93;. It then follows that any polynomial will have roots, either in the original field of its coefficients or in some extension field.

1995, Terence Jackson, From Polynomials to Sums of Squares, Taylor & Francis, page 56:

An extension field, by which we mean a bigger field containing F, is automatically a vector space over F. We call it a finite extension if it is a finite vector space. By the degree of a finite extension we mean its dimension as a vector space. One common way of obtaining extension fields is to adjoin an element to F: we say that K#61;F(#92;alpha) if K is the field consisting of all rational expressions formed using #92;alpha and elements of F.

1998, Neal Koblitz, Algebraic Aspects of Cryptography, Volume 3, Springer, page 53:

(algebra, field theory) A field L which contains a subfield K, called the base field, from which it is generated by adjoining extra elements.


Noetherian ring

Noetherian ring (plural Noetherian rings)

Noetherian rings (plural)
noetherian ring (alternative)

/nəˈtɛ.ɹi.ən ˈɹɪŋɡ/

Named after German mathematician Emmy Noether (1882–1935).

The central position occupied by Noetherian rings in commutative ring theory became evident from her^([Noether's]) work.

1986, Hideyuki Matsumura, translated by M. Reid, Commutative Ring Theory, Paperback edition, Cambridge University Press, published 1989, page ix:

In this chapter the focus moves from semiprime rings to general Noetherian rings, although it does concentrate on prime and semiprime ideals.

2000, John C. McConnell, James Christopher Robson, Lance W. Small, Noncommutative Noetherian Rings, 2nd edition, American Mathematical Society, page 97:

2004, K. R. Goodearl, Introduction to the Second Edition, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 2nd Edition, page viii,
During this same period, the explosive growth of the area of quantum groups provided a large new crop of noetherian rings to be analyzed, and thus gave major impetus to research in noetherian ring theory.

(algebra, ring theory) A ring which is either: (a) a commutative ring in which every ideal is finitely generated, or (b) a noncommutative ring that is both left-Noetherian (every left ideal is finitely generated) and right-Noetherian (every right ideal is finitely generated).


algebraic closure

algebraic closure (plural algebraic closures)

The calculation of these various Galois groups leads to a determination of the algebraic closures of the ground fields in the splitting fields of the corresponding vectorial polynomials.

1999, Shreeram S. Abhyankar, “Galois Theory of Semilinear Transformations”, in Helmut Voelklein, David Harbater, J. G. Thompson, Peter Müller, editors, Aspects of Galois Theory, Cambridge University Press, page 1:

It turns out that the algebraic closure #92;mathbf#123;Q#125;#92;mathrm#123;a#125;#95;p is not complete, so we shall consider its completion #92;mathbf#123;C#125;#95;p: This field turns out to be algebraically closed and is a natural domain for the study of "analytic functions."

2000, Alain M. Robert, A Course in p-adic Analysis, Springer, page 127:

While #92;Complex contains an algebraic closure of #92;mathbb#123;Q#125;, it is by no means the only algebraically closed field containing an algebraic closure of #92;mathbb#123;Q#125;. We denote by #92;mathbb#123;Q#125;#92;mathrm#123;alg#125; the algebraic closure of #92;mathbb#123;Q#125; in #92;Complex; this field is simply the subfield of #92;Complex consisting of algebraic numbers. The field #92;mathbb#123;Q#125;#92;mathrm#123;alg#125; is isomorphic, then, to any algebraic closure of #92;mathbb#123;Q#125;, but even knowing that it is unique up to isomorphism very likely leaves us no more familiar with #92;mathbb#123;Q#125;#92;mathrm#123;alg#125; than we were.

2004, John Swallow, Exploratory Galois Theory, Cambridge University Press, page 179:

(algebra, field theory, of a field F) A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G).

algebrallinen sulkeuma

field which is algebraically closed over another one


supercatenoid

LL-Q1860 (eng)-Flame, not lame-supercatenoid.wav

(algebra, geometry) a geometric surface constituting a transition between a cube (or similar surface) and a catenoid.


UFD

unique factorization domain
unique factorization domain#English

(algebra) Initialism of unique factorization domain.


unique factorization ring

unique factorization ring (plural unique factorization rings)

(algebra, ring theory) A ring in which every non-zero, non-unit (i.e., proper) element can be factored into a product of prime elements (not necessarily distinct), such that such factorization is unique up to permutation of the factors and the replacing of primes by their associate primes.


unique factorization domain

unique factorization domain (plural unique factorization domains)

unique factorization ring
unique factorization ring

(algebra, ring theory) A unique factorization ring which is also an integral domain.


principal ideal ring

principal ideal ring (plural principal ideal rings)

(algebra, ring theory) A commutative ring in which every ideal is a principal ideal.

(algebra, ring theory) A noncommutative ring in which every left ideal is a principal ideal and so is every right ideal.


commutative ring

commutative ring (plural commutative rings)

Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications.

1960, Oscar Zariski, Pierre Samuel, Commutative Algebra II, Springer, page 129:

As usual, it is simpler to begin by looking at a more general setting—in this case, commutative rings—before getting involved with polynomial rings. It turns out that the nature of the ideals in a commutative ring is important: for example, we have already seen that gcd's exist in PIDs, while this may not be true in other commutative rings.

2002, Joseph J. Rotman, Advanced Modern Algebra, 2nd edition, American Mathematical Society, page 295:

In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab#92;inP, it follows that a#92;inP or b#92;inP; equivalently, P is a prime ideal if and only if the factor ring R#47;P is a domain.

2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, page 47:

(algebra, ring theory) A ring whose multiplicative operation is commutative.

anello commutativo

a ring whose multiplicative operation is commutative


Dedekind domain

Dedekind domain (plural Dedekind domains)

Named after German mathematician Richard Dedekind (1831–1916).

It can be proved that a Dedekind domain (as defined above) is equivalent to an integral domain in which every proper fractional ideal is invertible.

In this chapter we shall study several of the important classes of rings which contain the class of Dedekind domains.

1971, Max D. Larsen, Paul J. McCarthy, Multiplicative Theory of Ideals, Elsevier (Academic Press), page 201:

As we can see every principal ideal domain is a Dedekind domain.

2007, Leonid Kurdachenko, Javier Otal, Igor Ya. Subbotin, Artinian Modules over Group Rings, Springer (Birkhäuser), page 55:

Let us recall some material about Dedekind domains from Chapters VIII and IX of Basic Algebra. A Dedekind domain is a Noetherian integral domain that is integrally closed and has the property that every nonzero prime ideal is maximal. Any Dedekind domain has unique factorization for its ideals.

2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 266:

(algebra, ring theory) An integral domain in which every proper ideal factors into a product of prime ideals which is unique (up to permutations).

anneau de Dedekind

integral domain whose prime ideals factorise uniquely


De Morgan algebra

De Morgan algebra (plural De Morgan algebras)

De Morgan algebras (plural)
de Morgan algebra (alternative)

Named after British mathematician and logician Augustus De Morgan (1806–1871). The notion was introduced by Grigore Moisil.

Finally it is shown that the compact elements in the congruence lattice of a De Morgan algebra form a Boolean sublattice.

1980, H. P. Sankappanavar, “A Characterization of Principal Congruences of De Morgan Algebras and its Applications”, in A. I. Arruda, R. Chuaqui, N. C. A. Da Costa, editors, Mathematical Logic in Latin America: Proceedings of the IV Latin American Symposium on Mathematical Logic, page 341:

By a topological de Morgan algebra we shall mean an abstract algebra (A,#92;land,#92;lor,#92;',l) where (A,#92;land,#92;lor,l) is a de Morgan algebra,

2000, Luo Congwen, Topological De Morgan Algebras and Kleene-Stone Algebras: The Journal of Fuzzy Mathematics, Volume 8, Pages 1-524, page 268:

If (L,#92;le,#123;⁻#125;) is a bounded distributive lattice with negation function (resp. a De Morgan algebra), then (L#92;langle#92;#33;#92;langleS#92;rangle#92;#33;#92;rangle,#92;le,#123;⁻#125;) constitutes also a bounded distributive lattice with negation function (resp. a De Morgan algebra); for every r#92;inL#92;langle#92;#33;#92;langleS#92;rangle#92;#33;#92;rangle its negation #92;overline#123;r#125;#92;inL#92;langle#92;#33;#92;langleS#92;rangle#92;#33;#92;rangle is defined by (#92;overline#123;r#125;,s)#61;#92;overline#123;(r,s)#125; for every s#92;inS.

2009, George Rahonis, “Chapter 12: Fuzzy Languages”, in Manfred Droste, Werner Kuich, Heiko Vogler, editors, Handbook of Weighted Automata, Springer, page 486:

distributive lattice
distributive lattice

(algebra, order theory) A bounded distributive lattice equipped with an involution (typically denoted ¬ or ~) which satisfies De Morgan's laws.

Augustus De Morgan
De Morgan algebra
Grigore Moisil


Kleene algebra

(algebra) A De Morgan algebra which also satisfies the inequation x∧∼x⩽y,∨∼y for all x and y, where "∼" here denotes the De Morgan involution.


ordered integral domain

ordered integral domain (plural ordered integral domains)

(algebra) An integral domain which has a subset whose elements are said to be "positive", such that this subset is closed under addition, closed under multiplication, and all elements of the integral domain satisfy a law of trichotomy; namely, that either that element is in the said subset, or it is the zero (additive identity), or its product with −1 (the additive inverse of the multiplicative identity) belongs to the said subset.


semijoin

LL-Q1860 (eng)-Flame, not lame-semijoin.wav

(algebra, databases) A type of join in relational algebra, denoted by ⋉ or ⋊.


maxterm

LL-Q1860 (eng)-Flame, not lame-maxterm.wav

All the maxterms in a product of maxterms should have the same variables, although each maxterm should differ from every other one by the pattern of complementation of those variables.

(algebra) A sum that is a logical OR of a set of variables where each individual variable only appears once in the sum, either in complemented or uncomplemented form, so that the value of the sum becomes 0.


Euclidean domain

Euclidean domain (plural Euclidean domains)

So called because for such domains it is possible to apply the Euclidean algorithm in order to obtain greatest common divisors.

(algebra) an integral domain in which division with remainder is possible


permutation group

permutation group (plural permutation groups)

In this chapter we shall be concerned with the relationship between permutation groups and graphs. We begin by explaining how a transitive permutation group may be represented graphically, and then we reverse the process, showing that a graph gives rise to a permutation group.

1979, Norman L. Biggs, A. T. White, Permutation Groups and Combinatorial Structures, page 80:

The Galois group G(L_f /C(x)) is called the monodromy group of f, denoted Mon(f), and viewed as a permutation group on the conjugates of y over C(x).

1996, Helmut Volklein, Groups as Galois Groups: An Introduction, page 47:

Now, groups are axiomatically defined, and the above concept is a permutation group, that is, a subgroup of the symmetric group.[…]The study of finite permutation groups is one of the oldest parts of group theory, motivated initially by its connection with solvability of equations.

2002, Peter J. Cameron, “B.5 Permutation Groups”, in Alexander V. Mikhalev, Günter F. Pilz, editors, The Concise Handbook of Algebra, page 86:

(algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition.

transformation group

group whose elements are permutations of a set


adfected

LL-Q1860 (eng)-Flame, not lame-adfected.wav

From the Latin adfectus, variant spelling of affectus (“endowed with, possessed of; affected; impaired, weakened”), whence affect and, indirectly, affected.

(algebra, archaic, of an equation) Having different powers of the unknown quantity in its various terms.


K-theory

From circa 1960. The K stands for German Klasse (“class”).
The theory developed out of algebraic geometry after the 1957 publication of work by German-born French mathematician Alexander Grothendieck.

algebraic K-theory

study of rings generated by vector bundles

K'''-theory as an independent discipline is a fairly new subject, only about 35 years old.

1994, Jonathan Rosenberg, Algebraic K-Theory and Its Applications, Springer, page 1:

K'''-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C^*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K'''-theory (for C^*-algebras) is a pair of functors, called K₀ and K₁, that to each C^*-algebra A associate two Abelian groups K₀(A) and K₁(A).

2000, M. Rørdam, F. Larsen, Flemming Larsen, N. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, page ix:

topological K-theory
topological K-theory

(uncountable, algebra, algebraic geometry, algebraic topology) The study of rings R generated by the set of vector bundles over some topological space or scheme;

The study of rings R generated by the set of vector bundles over some topological space or scheme;
(dated, obsolete) that part of algebraic topology comprising what is now called topological K-theory.


algebraic notation

algebraic notation (usually uncountable, plural algebraic notations)

LL-Q1860 (eng)-Flame, not lame-algebraic notation.wav

notação algébrica

a method for recording chess moves using letters and numbers

(algebra) A method for recording domain and range of a function.


seminormal

LL-Q1860 (eng)-Flame, not lame-seminormal.wav

(algebra, of a ring) Being a commutative reduced ring in which, whenever x, y satisfy x³=y², there is s with s²=x and s³=y.


dual number

If f(x) is a polynomial or a power series then its derivative can be obtained “automatically” using dual numbers as follows: f'(x)#61;#123;f(x#43;#92;epsilon)-f(x)#92;over#92;epsilon#125; where f(x#43;#92;epsilon) should be expanded using the algebra of dual numbers. The role of the ε is therein similar to that of an infinitesimal.

In the special case of the algebra #92;mathbf#123;D#125; of dual numbers we get the vertical tangent bundle V.

1993, Ivan Kolář, Peter W. Michor, Jan Slovák, Natural Operations in Differential Geometry, Springer, page 336:

These numbers are now called dual numbers (a term due to Study) and can be defined as expressions of the form a#43;b#92;epsilon, #92;epsilon²#61;0. When this is done as Kotel'nikov and Study showed, the dual distance between two points of this sphere representing two lines, forming an angle #92;varphi and having shortest distance d, is equal to the dual number #92;varphi#43;#92;epsilond, while the motions of Euclidean space are represented by rotations of the dual sphere, and consequently depend on three dual parameters.

1996, Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevich, translated by Roger Cooke, Mathematics of the 19th Century: Geometry, Analytic Function Theory, Birkhäuser, page 87:

Thus we investigate natural T-functions defined on T#42;T#123;#92;mathbf#123;D#125;#92;otimesA#125;M to determine all natural operators T#92;toTT#42;Tᴬ for #92;mathbf#123;D#125; denoting the algebra of dual numbers.

2002, Jiří Tomáš, “Some Classification Problems on Natural Bundles Related to Weil Bundles”, in Olga Gil-Medrano, Vicente Miquel, editors, Differential Geometry, Valencia 2001: Proceedings of the International Conference, World Scientific, page 297:

(algebra) An element of an algebra (the algebra of dual numbers) which includes the real numbers and an element ε which satisfies ε ≠ 0 and ε² = 0.


Eisenstein integer

Eisenstein integer (plural Eisenstein integers)

Named after Gotthold Eisenstein (1823–1852), German mathematician.

To divide an Eisenstein integer a#43;b#92;omega by another Eisenstein integer c#43;d#92;omega, notice that (c#43;d#92;omega)(c#43;d)(c#43;d#92;omega²)#61;c³#43;d³; accordingly multiply both denominator and numerator (of the division expressed as a fraction) by (c#43;d)(c#43;d#92;omega²), then simplify.

(algebra) A complex number of the form a+bω, where a and b are integers and ω is defined by the following two rules: (1) ω³=1 and (2) 1+ω+ω²=0; an element of the Euclidean domain ℤ[ω].


biquaternion

(algebra) Any of the numbers w+xi+yj+zk! where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group.


R-module

(algebra) The monoid action of a ring R on an abelian group.


skew field

skew fields (plural)
skewfield (alternative)

(algebra, dated) A ring in which every nonzero element has a multiplicative inverse; division ring


Newton's method

(algebra, calculus) A method for finding successively better approximations to the roots (or zeroes) of a real-valued function.


Coxeter-Dynkin diagram

Coxeter-Dynkin diagram (plural Coxeter-Dynkin diagrams)

After mathematicians H. S. M. Coxeter and Eugene Dynkin.

A Coxeter-Dynkin diagram encodes the information in a Coxeter matrix, which in turn encodes the presentation of a Coxeter group.

Vertices of a Coxeter-Dynkin diagram represent generators of the corresponding Coxeter group. The generators (algebraic) in turn correspond to the reflecting hyperplanes (geometric). A pair of vertices which are not linked by an edge correspond to a pair of commuting generators. An unnumbered edge between a pair of vertices means that the braid relation between the corresponding generators has length three (e.g., aba = bab if the generators are a and b). An edge numbered ≥4 means that the braid relation between the corresponding generators has a length equal to that number. For example, if the edge is numbered 4 then the braid relation is cdcd = dcdc if the generators are c and d. If a set of edges form a cycle then the Coxeter group can be shown to be infinite. If a tree in a Coxeter-Dynkin diagram has more than one numbered edge then the Coxeter group can be shown to be infinite. There are a few more such rules, which ensure that finite Coxeter groups have Coxeter-Dynkin diagrams with relatively simple shapes.

Let 𝒬 be an indecomposable root lattice and let Γ denote the Coxeter-Dynkin diagram of the underlying root system Δ.

1995 June, R. V. Moody, J. Patera, “Voronoi Domains and Dual Cells in the Generalized Kaleidoscope with Applications to Root and Weight Lattices”, in Canadian Journal of Mathematics, page 597:

There is a deep and only partially understood connection between the classification and structure of singularities and the Coxeter-Dynkin diagrams introduced by H. S .M. Coxeter for classification of reflection-generated groups and by E. B. Dynkin for classification of semisimple Lie algebras.

2000, Andrei Gabrielov, “Coxeter-Dynkin diagrams and singularities”, in Evgeniĭ Borisovich Dynkin, A. A. Yushkevich, Gary M. Seitz, A. L. Onishchik, editors, Selected Papers of E. B. Dynkin with Commentary, page 367:

For 3 ≤ n ≤ 5, we will use Eₙ to denote the Coxeter-Dynkin diagrams of types A₁ + A₂(N = 3), A₄(N = 4) and D₅(N = 5).

2012, Igor V. Dolgachev, Classical Algebraic Geometry: A Modern View, page 363:

(geometry, algebra) A graph with numerically labelled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).

Coxeter-Dynkin diagram
Eugene Dynkin
Harold Scott MacDonald Coxeter


carrier set

(algebra) the underlying set of an algebraic structure


Monster group

We next consider the possible #92;Z#95;n orbifoldings of #92;mathcal#123;V#125;² with respect to elements of the Monster group M [Tul].

1996, Michael P. Tuite, “Monstrous Moonshine and orbifolds”, in K.T. Arasu, J.F. Dillon, K. Harada, S. Sehgal, R. Solomon, editors, Groups, Difference Sets, and the Monster: Proceedings of a Special Research Quarter, Walter de Gruyter, page 443:

We note that c(1)#61;196883; this is not just a numerological coincidence, but follows from a branch of mathematics called moonshine theory which arises from string theory and relates the Monster group and modular functions.

2008, L. J. P. Kilford, Modular Forms: A Classical and Computational Introduction, Imperial College Press, page 106:

They were also crucial in understanding the Monster and Moonshine phenomena, which refers to the fact that the dimensions of the irreducible representations of the largest sporadic finite group, the monster group, show up in the q-expansion coefficients of the elliptic modular function j.

2014, Martin Schlichenmaier, Krichever-Novikov Type Algebras: Theory and Applications, Walter de Gruyter, page 340:

(algebra) The largest sporadic group, of order 2⁴⁶ · 3²⁰ · 5⁹ · 7⁶ · 11² · 13³ · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 (approximately 8 · 10⁵³), denoted M or F₁.


cosingular

LL-Q1860 (eng)-Flame, not lame-cosingular.wav

(algebra) Having the same singularities.


Galois group

Named after Évariste Galois, who first discovered them.

In this section, we show how to determine the Galois group and the roots of an irreducible polynomial of degree 2, 3, or 4.

1996, Patrick Morandi, Field and Galois Theory, Springer, page 123:

Let B be a Galois algebra over R with Galois group G, C the center of B, and K#61;#92;#123;g#92;inG#92;vertg(c)#61;c#92;#92;mathsf#123;for#92;each#125;#92;c#92;inC#92;#125;. A natural question is whether B is a central Galois algebra with Galois group K.

2004, George Szeto, Liangyong Xue, “On Central Galois Algebras of a Galois Algebra”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, CRC Press, page 493:

Our work actually gives an algorithm for computing Galois groups of polynomials f(X)#92;in#92;Q#91;X#93;:

2009, Steven H. Weintraub, Galois Theory, 2nd edition, Springer, page 128:

(algebra, Galois theory) The automorphism group of a Galois extension.

زمرة غالوا

specific group associated with a field extension


change sides, change signs

(algebra, mnemonic) An acronym to help remember that when a value is moved to the opposite side of the equals sign, then the operator is changed also.


synthetic division

synthetic division (countable and uncountable, plural synthetic divisions)

(algebra) A method of polynomial long division in which one does not write the variables and some of the calculations.

división sintética

a method of polynomial long division in which one does not write the variables and some of the calculations


presemifield

LL-Q1860 (eng)-Flame, not lame-presemifield.wav

multiplicative identity
multiplicative identity#English

(mathematics, algebra) A finite algebra satisfying all the axioms for a skew field except multiplicative associativity and the existence of a multiplicative identity.


Henselian

From Hensel (surname) + -ian; after German mathematician Kurt Hensel (1861–1941).

1989, N. Bourbaki, Commutative Algebra: Chapters 1-7, [1985, N. Bourbaki, Éléments de Mathématique Algébre Commutative 1-4 et 5-7, Masson], Springer, page 256,
A local ring satisfying conditions (H) and (C) is called Henselian. Every complete Hausdorff local ring is Henselian. If A is Henselian and B is a commutative A-algebra which is a local ring and a finitely generated A-module, then B is Henselian.

An analogous result holds for Henselian fields. Recall that a field K is said to be Henselian with respect to a valuation v if v has a unique extension (also denoted v) to every algebraic extension of K.[…]It follows that every algebraic extension of an^([sic]) Henselian field K is Henselian.
[…] Every valued field (K,v) has a minimal separable algebraic extension (K#95;v,v) which is Henselian. The valued field is unique up to a K-isomorphism and is called the Henselian closure of (K,v) [Ribenboim, p. 176].

2008, Michael D. Fried, Moshe Jarden, Field Arithmetic, 3rd edition, Springer, page 203:

Although the study of the definability of henselian valuations has a long history starting with J. Robinson, most of the results in this area were proven during the last few years.

2017, Arno Fehm, Franziska Jahnke, “Recent progress on definability of Henselian valuations”, in Fabrizio Broglia, Françoise Delon, Max Dickman, Danielle Gondard-Cozette, Victoria Ann Powers, editors, Ordered Algebraic Structures and Related Topics: International Conference, American Mathematical Society, page 136:

(algebra, of a ring or field) Which satisfies the criteria for (some formulation of) Hensel's lemma.


plethysm

LL-Q1860 (eng)-Flame, not lame-plethysm.wav

Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm.

2015, Thomas Kahle, Mateusz Michalek, “Obstructions to combinatorial formulas for plethysm”, in arXiv:

(algebra) A particular group-theoretic operation on a set of functions of a given symmetry type.


HDA

HDA (countable and uncountable, plural HDAs)

(algebra) Initialism of higher-dimensional algebra.


vertex form

vertex form (countable and uncountable, plural vertex forms)

Named because of its use in finding the vertex of a parabola: (h,k).

(mathematics, algebra) a quadratic equation in the form y=a(x-h)²+k


Galois field

Named after French mathematician Évariste Galois (1811–1832).

named-after

The Galois field #92;mathrm#123;GF#125;(pⁿ) has order pⁿ and characteristic p.

The Galois field #92;mathrm#123;GF#125;(pⁿ) is a finite extension of the Galois field #92;mathrm#123;GF#125;(p) and the degree of the extension is n.

The multiplicative subgroup of a Galois field is cyclic.

A Galois field #92;mathbb#123;F#125;#95;#123;pⁿ#125; is isomorphic to the quotient of the polynomial ring #92;mathbb#123;F#125;#95;p adjoin x over the ideal generated by a monic irreducible polynomial of degree n. Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: #92;mathbb#123;F#125;#95;#123;pⁿ#125;#92;cong#123;#92;mathbb#123;F#125;#95;p#91;x#93;#92;over(#92;hatf#95;n(x))#125;.

1958 [Chelsea Publishing Company], Hans J. Zassenhaus, The Theory of Groups, 2013, Dover, unnumbered page,
A field with a finite number of elements is called a Galois field.
The number of elements of the prime field k contained in a Galois field K is finite, and is therefore a natural prime p.

The case of most interest to us will be that in which F is a finite field, the Galois field GF(q) for some prime power q. If q is prime, this field is #92;mathbb#123;Z#125;#95;q, the integers 0,1,#92;dots,q-1 with arithmetic modulo q.

2001, Joseph E. Bonin, A Brief Introduction To Matroid Theory, retrieved 2016-05-05:

2006, Debojyoti Battacharya, Debdeep Mukhopadhyay, D. RoyChowdhury, A Cellular Automata Based Approach for Generation of Large Primitive Polynomial and Its Application to RS-Coded MPSK Modulation, Samira El Yacoubi, Bastien Chopard, Stefania Bandini (editors), Cellular Automata: 7th International Conference, Proceedings, Springer, LNCS 4173, page 204,
Generation of large primitive polynomial over a Galois field has been a topic of intense research over the years. The problem of finding a primitive polynomial over a Galois field of a large degree is computationaly expensive and there is no deterministic algorithm for the same.

(algebra) A finite field; a field that contains a finite number of elements.


symmetry group

[The] situation is more delicate in the case of higher dimensional symmetry groups; it is not in general possible to reduce the order of an equation invariant under an r-parameter symmetry group by r using only quadratures.

1993, Peter J. Olver, Applications of Lie Groups to Differential Equations, published 2000, page 76:

It is often assumed that the design unit inside each fundamental region (particularly for monotranslational symmetry groups p111 and p1a1 and ditranslational symmetry groups p1 and pg) is asymmetric.

2000, C. E. Horne, Geometric Symmetry in Patterns and Tilings, page 79:

The structure of the symmetry group of a molecule affects several spectroscopic aspects and vice versa. Thus, it is clearly important to know the possible symmetries of fullerenes and similar structures.
In this chapter, we study symmetry groups of fulleroids.

2010, Stanislav Jendrol, František Kardoš, “28: Symmetry of Fulleroids”, in Klaus D. Sattler, editor, Handbook of Nanophysics: Clusters and Fullerenes, pages 28–1:

(geometry, algebra, group theory) A group whose elements are all the transformations under which a given object remains invariant and whose group operation is function composition.

full symmetry group

emphasises the inclusion of orientation-reversing transformations


Coxeter diagram

en-noun

Coxeter diagram (plural Coxeter diagrams)

It suffices to make use of the tables above and observe that the Coxeter diagram of a direct sum of C-matrices is the disconnected union of the Coxeter diagrams of the summands.
For the description of C⁻-matrices satisfying condition (2) we shall use Coxeter diagrams with the additional stipulation that in case aᵢⱼ < −1 the vertices vᵢ and vⱼ are joined by a dotted line with index −aᵢⱼ. By the Coxeter diagram of a Coxeter polyhedron and a lattice generated by reflections we mean the Coxeter diagram of the corresponding C⁻-matrix.

1991, Gregori A. Margulis, Discrete Subgroups of Semisimple Lie Groups, page 358:

An automorphism of a Coxeter diagram M leads in a natural way to a Coxeter subgroup of the Coxeter group of type M. We introduce admissible partitions of Coxeter diagrams in order to generalize this situation.

1993, B. Mühlherr, “Coxeter groups in Coxeter groups”, in Albrecht Beutelspacher, F. Buekenhout, J. Doyen, F. de Clerck, J. A. Thas, J. W. P. Hirschfeld, editors, Finite Geometries and Combinatorics, 2nd International Conference, page 277:

The Coxeter diagrams of type E₇ and E₈ have no nontrivial automorphisms, so σ₀ is trivial in those cases.[…]The Coxeter diagram of type F₄ has a unique nontrivial automorphism, but σ₀ is trivial in this case. One can see this by using the Dynkin diagram instead of the Coxeter diagram and noting that it does not have any nontrivial automorphisms.

2008, Peter Abramenko, Kenneth S. Brown, Buildings: Theory and Applications, page 259:

Coxeter-Dynkin diagram
Coxeter-Dynkin diagram

(geometry, algebra) A Coxeter-Dynkin diagram.


semiartinian

LL-Q1860 (eng)-Flame, not lame-semiartinian.wav

(algebra) Of a module _RM: such that, for every M→N epimorphism, where N≠0, the socle of N is essential in N.


inverse semigroup

inverse semigroup (plural inverse semigroups)

Examples of regular semigroups include any band, inverse semigroup or completely regular semigroup (see sections A. 2, A. 11, and A. 10); in particular, any group is a regular semigroup.

2002, Peter G. Trotter, “A.9: Regular Semigroups”, in Aleksandr Vasilʹevich Mikhalev, Günter Pilz, editors, The Concise Handbook of Algebra, page 35:

Inverse semigroups form a class of semigroups which are closest (in some sense) to groups.

2008, Olexandr Ganyushkin, Volodymyr Mazorchuk, Classical Finite Transformation Semigroups: An Introduction, page v:

Inverse semigroups are central to modern semigroup theory: arguably, they form the most-studied class of semigroups.

2014, Christopher Hollings, Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, page 249:

(algebra, group theory) A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy.

inverse Halbgruppe

semigroup in which every element has an inverse


Kantor-Koecher-Tits construction

Introduced by Jacques Tits (1962), Kantor (1964), and Koecher (1967).

(algebra) A method of constructing a Lie algebra from a Jordan algebra.


bicomplex number

bicomplex number (plural bicomplex numbers)

(algebra) A pair (w,z) of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^*=(w,-z), and the product of two bicomplex numbers as (u,v)(w,z)=(uw-vz,uz+vw).


adic

(algebra, of topological ring) Carrying the I-adic topology for some ideal I of the ring.


bispace

LL-Q1860 (eng)-Flame, not lame-bispace.wav

(algebra) A set with both left and right morphisms.


Dixmier conjecture

(algebra) The conjecture that any endomorphism of a Weyl algebra is an automorphism.


Weyl algebra

Named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics.

differential operator
differential operator

(algebra) The ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form f_m(X)∂_Xᵐ+f_m-1(X)∂_Xᵐ⁻¹+⋯+f_1(X)∂_X+f_0(X).


Zhegalkin polynomial

Zhegalkin polynomial (plural Zhegalkin polynomials)

Named after the Russian mathematician Ivan Ivanovich Zhegalkin (1869–1947), who introduced them in 1927.

(logic, algebra) A Boolean-valued polynomial formed as a XOR-sum (or "ring sum") of conjunctions.

Ivan Ivanovich Zhegalkin
Zhegalkin polynomial


ring sum normal form

"Ring sum" refers to a XOR-sum of monomials, where XOR is the additive operator of a Boolean ring.

(algebra, logic) The form of a Boolean formula expressed using only the operators and constants of a Boolean ring (XOR, AND, 0, 1), some variables and possibly also coefficients, and without using any parentheses.


algebraic normal form

"Algebraic" would refer to the fact that a two-element Galois field, 𝔽₂ adjoined with some variables forms a polynomial algebra, 𝔽₂[x_1,...,x_n].

It is well known that every (n,1)-function f can be written uniquely as a multilinear polynomial over #92;mathbb#123;F#125;#95;2
#92;qquad#92;qquad#92;qquad#92;qquad#92;qquadf(x#95;1,...,x#95;n)#61;#92;sum#95;#123;X#92;subseteq#91;n#93;#125;#92;alpha#95;X#92;prod#95;#123;i#92;inX#125;x#95;i.
This polynomial is called the Zhegalkin polynomial or the algebraic normal form (ANF) of f.

2015 September 22, Magnus Gausdal Find, Joan Boyar, “Multiplicative Complexity of Vector Valued Boolean Functions”, in arxiv.org, retrieved 2017-08-10:

ring sum normal form
ring sum normal form


Boolean derivative

Boolean derivative (plural Boolean derivatives)

Named after George Boole (1815–1864), an English mathematician, educator, philosopher and logician.

(algebra) With respect to a Boolean variable x_i: The XOR-sum of the positive and negative Shannon cofactors of the given derived function with respect to x_i. In symbols: ∂f/∂x_i(⃑x)=f_x_i(⃑x)⊕f_̄x_i(⃑x).


underlying set

(algebra) A set upon which some operations are defined, thereby yielding an algebraic structure.


formal power series

formal power series (plural formal power series)

1995 Alberto Bertoni, Massimiliano Goldwurm, Giancarlo Mauri, Nicoletta Sabatini, Chapter 5: Counting Techniques for Inclusion, Equivalence, and Membership Problems, Volker Diekert, Grzegorz Rozenberg, The Book of Traces, World Scientific, page 138,
Moreover, it is useful to observe that in this case the definition of rational formal power series can be simplified: a f.p.s. r∈ℚ⟨!⟨z⟩!⟩ is rational if and only if r is the quotient of two polynomials.

1997, Greg Marks, Direct Product and Power Series Formations over 2-Primal Rings, Surender Kumar Jain, S. Tariq Rizvi, Advances in Ring Theory, Springer, page 239,
We also show that the ring of formal power series over a 2-primal ring (or even a ring satisfying (PS I)) need not be 2-primal.

This chapter gives a rigorous development of the algebraic properties of formal power series. Our goal is to extend the familiar operations on polynomial functions (like addition, multiplication, composition, and differentiation) to the setting of formal power series. In certain situations we will even be able to define infinite sums and products of formal power series.[…]In combinatorics, it usually suffices to consider formal power series whose coefficients are integers, rational numbers, or complex numbers.

2011, Nicholas Loehr, Bijective Combinatorics, Taylor & Francis (CRC Press), page 243:

(mathematics, algebra) Any finite or infinite series of the form a_0+a_1x+a_2x²…=∑ᵢa_ixⁱ, where the aᵢ are numbers, but it is understood that no value is assigned to x.


quadratic form

Quadratic form
Quadratic form (statistics)

Historically, quadratic form theory has been treated as a rich but misunderstood uncle.

1980, John Gregory, Quadratic Form Theory and Differential Equations, Academic Press, page ix:

2004, Nikita A. Karpenko, Izhboldin's Results on Stably Birational of Equivalence Quadrics, Oleg T. Izhboldin, Geometric Methods in the Algebraic Theory of Quadratic Forms, Springer, Lecture Notes in Mathematics 1835, page 156,
We claim that this quadratic form is isotropic, and this gives what we need according to [2, Proposition 4.4].

2009, R. Parimala, V. Suresh, J.-P. Tignol, On the Pfister Number of Quadratic Forms, Ricardo Baeza, Quadratic Forms—Algebra, Arithmetic, and Geometry, American Mathematical Society, page 327,
The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of −1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n − 2 terms and not less. The number of Pfister forms required to express a quadratic form of degree 6 with trivial discriminant is determined in various cases.

The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange's theorem that every positive integer is a sum of four squares, was continued in the 19th century by Dirichlet, Hermite, H. J. S. Smith, Minkowski and others.

2009, W. A. Coppel, Number Theory: An Introduction to Mathematics, 2nd edition, Springer, page 291:

homogeneous polynomial
homogeneous polynomial

(mathematics, number theory, algebra) A homogeneous polynomial of degree 2 in a given number of variables.


directed set

(algebra, set theory) A set equipped with a preorder such that for any two elements a and b there is an element c such that a ≤ c and b ≤ c.


field of quotients

field of quotients (plural fields of quotients)

(algebra) A field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so: (a,b)+(a',b')=(ab'+a'b,bb'), the multiplicative operator is defined coordinate-wise, the zero is (0,1), the unity is (1,1), the additive inverse of (a,b) is (-a,b), equivalence is defined like so: (a,b)≡(a',b') if and only if ab'=a'b, and multiplicative inverse of a non-zero–equivalent element (a,b) is (b,a).


polynomial form

polynomial form (plural polynomial forms)

(algebra) A linear combination of powers of an indeterminate (or products of powers of more than one indeterminate), with coefficients belonging to an integral domain or a field. (The indeterminate is thought of as an element extraneous to the set of coefficients, instead of as a variable element of it (as in the case of polynomial functions), just as, say, the square root of negative one is an element extraneous to the set of integers when it is adjoined to them to form the domain of Gaussian integers. The indeterminate forms a free commutative monoid, to which all powers of it belong, and the unity of it can also show up implicitly in the constant term of a polynomial form.)


representation theory

representation theory (countable and uncountable, plural representation theories)

1989 [Clarendon Press], Robert J. Baston, Michael G. Eastwood, The Penrose Transform: Its Interaction with Representation Theory, 2016 Dover edition.

1998, A. Broer, A. Daigneault, Representation Theories and Algebraic Geometry, Springer:

2005, Andrea Brini, Francesco Regonati, Antonio Teolis, Combinatorics and Representation Theory of Lie Superalgebras over Letterplace Superalgebras, Hongbo Li, Peter J. Olver, Gerald Sommer (editors), Computer Algebra and Geometric Algebra with Applications: 6th International Workshop IWWM and International Workshop GIAE, Springer, LNCS 3519, page 239,
As an application, we describe in detail the way to specialize general results to the representation theory of the symmetric group and to classical invariant theory.

From his numerous contributions through research articles, books, and conference volumes we just mention his^([Vlastimil Dlab's]) work on the representation theory of hereditary algebras in the mid-seventies of the last century, joint with C.M. Ringel, laying the foundations of, in particular, tame hereditary representation theory over arbitrary fields, a theory still at the centre of interest.

2005, Ragnar-Olaf Buchweitz, Helmut Lenzing, Representations of Algebras and Related Topics, American Mathematical Society, page ix:

linear transformation
linear transformation

(mathematics, algebra) A branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.


free Boolean algebra

free Boolean algebra (plural free Boolean algebras)

(algebra) A field of sets whose elements are equivalent to Boolean formulas (or, perhaps more precisely, equivalence classes of Boolean formulas). Starting with a set of n variables which are independent of each other and are called generators, the power set of this set has 2ⁿmembers which may be called atoms and are valuations of the n variables: a valuation can be considered to be a set of variables which are "true" under that valuation, or a conjunction of generators (such that variables not included in that set are included in negated form in the equivalent conjunction). Then the power set of the set of atoms yields a set of 2^(2ⁿ) members which are the elements of the said field of sets. These elements correspond to Boolean formulas: a formula can be considered to be a set of valuations which make the formula true, or a linear combination (i.e., a disjunction) of atoms.


free abelian group

free abelian group (plural free abelian groups)

A free abelian group of rank n is isomorphic to ℤ⊕ℤ⊕...⊕ℤ=⨁ⁿℤ, where the ring of integers ℤ occurs n times as the summand. The rank of a free abelian group is the cardinality of its basis. The basis of a free abelian group is a subset of it such that any element of it can be expressed as a finite linear combination of elements of such basis, with the coefficients being integers. (For an element a of a free abelian group, 1a = a, 2a = a + a, 3a = a + a + a, etc., and 0a = 0, (−1)a = −a, (−2)a = −a + −a, (−3)a = −a + −a + −a, etc.)

(algebra) A free module over the ring of integers.


cyclotomic polynomial

cyclotomic polynomial (plural cyclotomic polynomials)

For a prime number p, the pᵗʰ cyclotomic polynomial is #123;xᵖ-1#92;overx-1#125;#61;x#123;p-1#125;#43;x#123;p-2#125;#43;...#43;x²#43;x#43;1.

Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing x with x#43;1.

Euler's totient function
Euler's totient function

(algebra) For a positive integer n, a polynomial whose roots are the primitive nᵗʰ roots of unity, so that its degree is Euler's totient function of n. That is, letting ζₙ=e^(i 2π/n) be the first primitive nᵗʰ root of unity, then Φₙ(x)=∏_( stackrel )1<m<ngcd (n,m)=1(x-ζₙᵐ) is the nᵗʰ such polynomial.


prime ideal

By analogy with the notion of prime number in number theory.

Given a prime number p, there is only a finite number of prime ideals #92;mathfrak#123;p#125; in #92;mathfrak#123;o#125; such that #92;mathfrak#123;p#125;#92;capJ#61;p (they are the prime ideals of #92;mathfrak#123;o#125;p).

1960, [Van Nostrand], Oscar Zariski, Pierre Samuel, Commutative Algebra, volume II, Springer, published 1975, page 39:

1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 2003, Springer, page 189,
In the rings studied in Section 17.4 a nonzero prime ideal is divisible only by itself and by o on the basis of Axiom II; thus, in that section there are no lower prime ideals but o. Since every ideal a ne o is divisible by a prime ideal distinct from o (proof: from among all the divisors of a distinct from o choose a maximal one; since this ideal is maximal it is also prime), it follows that a cannot be quasi-equal to o.

In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab#92;inP, it follows that a#92;inP or b#92;inP; equivalently, P is a prime ideal if and only if the factor ring R#47;P is a domain.

2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47:

(algebra, ring theory) Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I.


free monoid

(algebra) A monoid whose underlying set is the Kleene closure of some set of generators, and whose operator is concatenation.


quotient ring

The third section covers a construct similar to the ring S°R — the maximal quotient ring, which exists for any ring. (When R is nonsingular, the maximal quotient ring is exactly S°R.) Finally, Section D provides an answer to the question of which right and left nonsingular rings have coinciding maximal right and left quotient rings.

1976, Kenneth Goodearl, Ring Theory: Nonsingular Rings and Modules, CRC Press, page 39:

On the other hand if already every element of R is either invertible or a zerodivisor, then R is its own classical quotient ring.

2006, Peter A. Linnell, “Noncommutative localization in group rings”, in Andrew Ranicki, editor, Noncommutative Localization in Algebra and Topology, Cambridge University Press, page 42:

2. An ideal P of the ring R is prime if and only if the quotient ring R/P is a domain.

2012, Oleg A. Logachev, A. A. Salnikov, V. V. Yashchenko, translated by Svetla Nikova, Boolean Functions in Coding Theory and Cryptography, American Mathematical Society, page 10:

(algebra, ring theory) For a given ring R and ideal I contained in R, another ring, denoted R / I, whose elements are the cosets of I in R.

anneau quotient

ring whose elements are the cosets of an ideal


fractional ideal

fractional ideal (plural fractional ideals)

Products of fractional ideals are again fractional ideals, since if A#92;alpha#92;subseteqR and B#92;beta#92;subseteqR, then (AB)(#92;alpha#92;beta)#92;subseteqR.

1994, I. Martin Isaacs, Algebra: A Graduate Course, American Mathematical Society, page 476:

Theorem 5 The non-zero fractional ideals of a Dedekind domain form a multiplicative group.

2001, H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, page 10:

2008, Jan Hendrik Bruinier, Hilbert Modular Forms and Their Applications, Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, Don Zagier (editors), The 1-2-3 of Modular Forms: Lectures at a Summer School, Springer, page 106,
A fractional ideal of F is a finitely generated 𝒪_F-submodule of F. Fractional ideals form a group together with the ideal multiplication. The neutral element is 𝒪_F and the inverse of a fractional ideal a⊂F is
a⁻¹=x∈F;xa⊂𝒪_F.
[…] Two fractional ideals a,b are called equivalent, if there is a r∈F such that a=rb.

(algebra, ring theory) Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R.

R-submodule of Frac(R) such that for some nonzero r∈R


field of fractions

field of fractions (plural fields of fractions)

1971 [Wadsworth Publishing], Allan Clark, Elements of Abstract Algebra, 1984, Dover, page 175,
The general construction of the field of fractions ℚ_R out of R is an exact parallel of the construction of the field of rational numbers ℚ out of the ring of integers ℤ.

1989, Nicolas Bourbaki, Commutative Algebra: Chapters 1-7, [1985, Éléments de Mathématique Algèbre Commutative, 1-4 et 5-7, Masson], Springer, page 535,
In this no., A and B denote two integrally closed Noetherian domains such that A ⊂ B and B is a finitely generated A-module and K and L the fields of fractions of A and B respectively.

We denote by #92;mathcal#123;A#125;(#92;mathbb#123;K#125;) the #92;mathbb#123;K#125;-algebra of entire functions in #92;mathbb#123;K#125; i.e. the set of power series with coefficients in #92;mathbb#123;K#125; converging in all #92;mathbb#123;K#125; and we denote by #92;mathcal#123;M#125;(#92;mathbb#123;K#125;) the field of meromorphic functions in #92;mathbb#123;K#125;, i.e. the field of fractions of #92;mathcal#123;A#125;(#92;mathbb#123;K#125;).

2013, Jean-Paul Bézivin, Kamal Boussaf, Alain Escassut, “Some old and new results on the zeros of the derivative of a p-adic meromorphic function”, in Khodr Shamseddine, editor, Advances in Ultrametric Analysis: 12th International Conference on p-adic Functional Analysis, American Mathematical Society, page 23:

(algebra, ring theory) The smallest field in which a given ring can be embedded.


factor ideal

1965 [Prentice-Hall], Seth Warner, Modern Algebra, 1990, Dover, page 651,
Consequently, we may speak of the sequence of invariant factor ideals of a finitely generated unitary module over a principal ideal domain A.

1985 [W. H. Freeman and Company], Nathan Jacobson, Basic Algebra I, 2nd Edition, Dover, 2009, page 193,
We shall now call the sequence of order ideals, ann z₁, ann z₂, ..., whose uniqueness has just been proved, the invariant factor ideals of the module M.

Recall that a factor ideal I of a half-shell A comes from a factor congruence θ of A where I = 0/θ. Factor ideals and congruences will be needed whenever we want to show that we have embedded a half-shell into a Baer–Stone half-shell.[…]
2.1 Definition. Remember that a half-shell A is called a Baer–Stone half-shell if the right and left annihilators of every single element of A are factor ideals. It is called finitely Baer-Stone if the annihilators on both sides of any finite subset of A are factor ideals.

2012, Arthur Knoebel, Sheaves of Algebras over Boolean Spaces, Springer (Birkhäeser), page 246:

(algebra, ring theory) For given ring R, any ideal I such that Q = R / I, the set of cosets of elements of I in R, is a ring (the quotient ring of I in R).


factor out

en-verb

factor out (third-person singular simple present factors out, present participle factoring out, simple past and past participle factored out)

factors out (present singular third-person)
factoring out (participle present)
factored out (participle past)
factored out (past)

(transitive, algebra) To isolate a common factor from an expression.

ottaa yhteinen tekijä

(algebra) to isolate a common factor from an expression


simple ring

Theorem 2. A semi-simple ring #92;mathfrak#123;A#125; satisfying the minimum condition can be decomposed in only one way as a direct sum of ideals which are simple rings.

1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page 43:

By theorem 7.7.1 any field is a simple ring.

1969, Frederick Michael Hall, An Introduction to Abstract Algebra, volume 2, Cambridge University Press, page 195:

A field is clearly a simple ring. Indeed, a commutative simple ring with unity must be a field (Problem 1, Section 1). An example of a noncommutative simple ring is Fₙ, the n × n matrix ring over a field F, n > 1.

1994, P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul, Basic Abstract Algebra, 2nd edition, Cambridge University Press, page 204:

Since M#95;n(D) has no nonzero proper two-sided ideals, it follows from the above discussion that M#95;n(D) is a left simple ring. We shall see that every simple ring is isomorphic to M#95;n(D) for some n, and for some division ring D.

2017, Ramji Lal, Algebra 2: Linear Algebra, Galois Theory, Representation theory, Group Extensions and Schur Multiplier, Springer, page 335:

(algebra, ring theory) A ring that contains no nontrivial ideals (i.e., no (two-sided) ideals other than the zero ideal and the ring itself).

anneau simple

ring that contains no ideals other than the zero ideal and the ring itself


simple algebra

1945 [McGraw-Hill], E. T. Bell, 2nd Edition, 1992, Dover, The Development of Mathematics, page 252,
A semi-simple algebra over F is either simple or the direct sum of simple algebras over F.

1989 [W. H Freeman & Company], Nathan Jacobson, Basic Algebra II, 2nd Edition, 2009, Dover, page 215,
If A is a finite dimensional simple algebra over F, then A≅M_n(Δ) where Δ is a finite dimensional division algebra over Δ.

One proves that if #92;mathcal#123;L#125; is a simple algebra over #92;mathbb#123;R#125;, then L#95;C is either a simple algebra over #92;mathbb#123;C#125;, or a direct sum of two isomorphic simple algebras. Thus the problem of studying simple algebras over #92;mathbb#123;R#125; reduces to a similar problem over #92;mathbb#123;C#125;.

2005, Igor R. Shafarevich, translated by Miles Reid, Basic Notions of Algebra, Springer, page 198:

(algebra) An algebra that contains no nontrivial proper (two-sided) ideals and whose multiplication operation is not zero (i.e., there exist a and b such that ab ≠ 0).

algebra semplice

algebra that contains no nontrivial proper ideals and whose multiplication operation is not zero


prime ring

A ring is called a prime ring if the product of nonzero ideals in it remains nonzero. It is obvious that a prime ring is necessarily semi-prime.

1969, Taita Journal of Mathematics, volumes 1-2, page 56:

The ring R is said to be prime if for all nonzero ideals A, B of R we have AB≠0. An ideal P of R is called a prime ideal if R/P is a prime ring. Prime rings and prime ideals are important building blocks in noncommutative ring theory.

1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products, Elsevier (North-Holland), page 223:

The so-called extended centroid of a prime ring, i.e., a field defined as the center of the Martindale ring of quotients, will enable us to extend a part of the theory of central simple algebras to general prime rings.

2014, Matej Brešar, Introduction to Noncommutative Algebra, Springer, page 163:

(algebra, ring theory) Any nonzero ring R such that for any two (two-sided) ideals P and Q in R, the product PQ = 0 (the zero ideal) if and only if P = 0 or Q = 0.

Moreover, the image of φ_A is the smallest subring of A, in the sense that it is contained in any subring of A, and it is called the prime ring of A.

2012, Sebastian Xambo-Descamps, Block Error-Correcting Codes: A Computational Primer, Springer Science & Business Media, page 110:

(algebra, ring theory) Synonym of prime subring.

The image of ℤ or ℤ/mℤ, respectively, in R as described in the above proposition obviously consists of all sums n · 1 in R, where n ∈ ℤ. It is also called the prime ring of R. R itself is called a prime ring if it equals its own prime ring. If p is a prime number, then the field ℤ/pℤ is a also called the prime field of characteristic p.

2012, Thomas Becker, Volker Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra, Springer Science & Business Media, page 50:

Theorem 3.6.3. If R is a prime ring of characteristic zero then R is isomorphic to ℤ. If R is a prime ring of characteristic n then R is isomorphic to ℤₙ.

2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra: From Rings, Numbers, Groups, and Fields to Polynomials and Galois Theory, JHU Press, page 81:

(algebra, ring theory, uncommon) A ring which is equal to its own prime subring.


inner automorphism

en-noun

inner automorphism (plural inner automorphisms)

(algebra) An automorphism which is a conjugation (by a fixed element).


outer automorphism

outer automorphism (plural outer automorphisms)

(algebra) An automorphism which is not an inner automorphism.


maximal ideal

Denote the minimal prime ideal in (b) by J#95;#92;mathsf#123;max#125;(#92;lambda), the unique maximal ideal of infinitesimal character #92;lambda.

1994, William M. McGovern, Completely Prime Maximal Ideals and Quantization, American Mathematical Society, page 15:

2004, Ayman Badawi, Abstract Algebra Manual: Problems and Solutions, Nova Science, 2nd Edition, page 87,
Let S be the set of all prime ideals of B, and H be the set of all maximal ideals of B.

In particular, a prime ideal #92;mathfrak#123;p#125;#92;subsetA is a closed point of #92;mathrm#123;Spec#125;#92;A if and only if #92;mathfrak#123;p#125; is a maximal ideal.

2013, Igor R. Shafarevich, Basic Algebraic Geometry 2: Schemes and Complex Manifolds, 3rd edition, Springer, page 11:

(algebra, ring theory) An ideal which cannot be made any larger (by adjoining any element to it) without making it improper (i.e., equal to the whole of the containing algebraic structure).


primary ideal

The prime and primary ideals play roles which are (very roughly) similar to those played by prime numbers and by prime.power numbers in elementary arithmetic.

1953, D. G. Northcott, Ideal Theory, Cambridge University Press, page 10:

1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, page 189,
Thus all higher primary ideals are symbolic powers of higher prime ideals.
Prüfer has called the ideals a with the property a* = a v-ideals. The integral v-ideals are just those in whose primary ideal decomposition only higher primary ideals occur.

A primary ideal has a prime ideal as radical, so its corresponding algebraic set is irreducible. Primary ideals can, however, have multiplicity, so they give a finer description of the solution set.

1997, Ralf Fröberg, An Introduction to Gröbner Bases, John Wiley & Sons, page 71:

(algebra, ring theory) Given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0.


conjugacy class

conjugacy class (plural conjugacy classes)

If a group acts on itself through conjugation, then its orbits are its conjugacy classes and its stabilizer subgroups are its centralizers.

equivalence relation
equivalence relation

(algebra) A subset of a group which is an equivalence class in the quotient set of the group divided by conjugation as equivalence relation.


minimal ideal

Of particular interest are the primitive rings with minimal ideals, algebraic algebras and algebras with a polynomial identity.

1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page vii:

Theorem 12. A semisimple ring R has only a finite number of minimal ideals and is their direct sum. Moreover, each minimal ideal is a simple ring.

1970, Mary W. Gray, A Radical Approach to Algebra, Addison-Wesley, page 33:

Note that if a semigroup has a zero, then its minimal ideal is {0}. In this context, the notion of minimal ideal is not so useful, and so we introduce the notion of a 0-minimal ideal.

2009, John Rhodes, Benjamin Steinberg, The q-theory of Finite Semigroups, Springer, page 596:

(algebra, ring theory) A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal.