判別式

/pʰän⁵¹ pi̯ɛ³⁵ ʂʐ̩⁵¹/

/pʰuːn³³ piːt̚² sɪk̚⁵/

/pʰän⁵¹ pi̯ɛ³⁵ ʂʐ̩⁵¹/

/pʰuːn³³ piːt̚² sɪk̚⁵/

(algebra) discriminant (function of a polynomial's coefficients)


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.


方程組

/fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵ t͡su²¹⁴⁻²¹⁽⁴⁾/

/fɔːŋ⁵⁵ t͡sʰɪŋ²¹ t͡sou̯³⁵/

/fɑŋ⁵⁵ ʈ͡ʂʰɤŋ³⁵ t͡su²¹⁴⁻²¹⁽⁴⁾/

/fɔːŋ⁵⁵ t͡sʰɪŋ²¹ t͡sou̯³⁵/

simultaneous equations
simultaneous equations


聯立

(algebra) to list (two or more equations) together and solve simultaneously


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).


閉包

/pi⁵¹ pɑʊ̯⁵⁵/

/pi⁵¹ pɑʊ̯⁵⁵/

(programming, algebra, topology) closure


nulelement

head

nulelement n (singular definite nulelementet, plural indefinite nulelementer)

nulelementet (definite singular)
nulelementer (indefinite plural)
nulelement (indefinite nominative singular)
nulelementet (definite nominative singular)
nulelementer (indefinite nominative plural)
nulelementerne (definite nominative plural)
nulelements (genitive indefinite singular)
nulelementets (definite genitive singular)
nulelementers (genitive indefinite plural)
nulelementernes (definite genitive plural)

(algebra) in a set, an element 0 such that, for a given addition + and an element a, a + 0 = 0 + a = a


etelement

etelement n (singular definite etelementet, plural indefinite etelementer)

etelementet (definite singular)
etelementer (indefinite plural)
etelement (indefinite nominative singular)
etelementet (definite nominative singular)
etelementer (indefinite nominative plural)
etelementerne (definite nominative plural)
etelements (genitive indefinite singular)
etelementets (definite genitive singular)
etelementers (genitive indefinite plural)
etelementernes (definite genitive plural)
ételement (alternative)

(algebra) identity element; in a set, an element 1 such that, for a multiplication * and an element a, a * 1 = 1 * a = a


多項式

多項式 (canonical)
takōshiki (romanization)


Zhegalkin polynomial

en-noun

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


बहुपद

bahupad (romanization)
बहुपद (direct singular)
बहुपद (direct plural)
बहुपद (oblique singular)
बहुपदों (oblique plural)
बहुपद (singular vocative)
बहुपदो (plural vocative)

Borrowed from Sanskrit बहुपद (bahupada, “many-footed”).

bor+

द्विपद बहुपद


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

en-noun

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).


qüvvət

az-noun

qüvvət (definite accusative qüvvəti, plural qüvvətlər)

qüvvəti (accusative definite)
qüvvətlər (plural)
qüvvət (nominative singular)
qüvvətlər (nominative plural)
qüvvəti (accusative definite singular)
qüvvətləri (accusative definite plural)
qüvvətə (dative singular)
qüvvətlərə (dative plural)
qüvvətdə (locative singular)
qüvvətlərdə (locative plural)
qüvvətdən (ablative singular)
qüvvətlərdən (ablative plural)
qüvvətin (definite genitive singular)
qüvvətlərin (definite genitive plural)
qüvvətim (first-person nominative possessive singular)
qüvvətlərim (first-person nominative possessive singular)
qüvvətin (nominative possessive second-person singular)
qüvvətlərin (nominative possessive second-person singular)
qüvvəti (nominative possessive singular third-person)
qüvvətləri (nominative possessive singular third-person)
qüvvətimiz (first-person nominative plural possessive)
qüvvətlərimiz (first-person nominative plural possessive)
qüvvətiniz (nominative plural possessive second-person)
qüvvətləriniz (nominative plural possessive second-person)
qüvvəti (nominative plural possessive third-person)
qüvvətləri (nominative plural possessive third-person)
qüvvətimi (accusative first-person possessive singular)
qüvvətlərimi (accusative first-person possessive singular)
qüvvətini (accusative possessive second-person singular)
qüvvətlərini (accusative possessive second-person singular)
qüvvətini (accusative possessive singular third-person)
qüvvətlərini (accusative possessive singular third-person)
qüvvətimizi (accusative first-person plural possessive)
qüvvətlərimizi (accusative first-person plural possessive)
qüvvətinizi (accusative plural possessive second-person)
qüvvətlərinizi (accusative plural possessive second-person)
qüvvətini (accusative plural possessive third-person)
qüvvətlərini (accusative plural possessive third-person)
qüvvətimə (dative first-person possessive singular)
qüvvətlərimə (dative first-person possessive singular)
qüvvətinə (dative possessive second-person singular)
qüvvətlərinə (dative possessive second-person singular)
qüvvətinə (dative possessive singular third-person)
qüvvətlərinə (dative possessive singular third-person)
qüvvətimizə (dative first-person plural possessive)
qüvvətlərimizə (dative first-person plural possessive)
qüvvətinizə (dative plural possessive second-person)
qüvvətlərinizə (dative plural possessive second-person)
qüvvətinə (dative plural possessive third-person)
qüvvətlərinə (dative plural possessive third-person)
qüvvətimdə (first-person locative possessive singular)
qüvvətlərimdə (first-person locative possessive singular)
qüvvətində (locative possessive second-person singular)
qüvvətlərində (locative possessive second-person singular)
qüvvətində (locative possessive singular third-person)
qüvvətlərində (locative possessive singular third-person)
qüvvətimizdə (first-person locative plural possessive)
qüvvətlərimizdə (first-person locative plural possessive)
qüvvətinizdə (locative plural possessive second-person)
qüvvətlərinizdə (locative plural possessive second-person)
qüvvətində (locative plural possessive third-person)
qüvvətlərində (locative plural possessive third-person)
qüvvətimdən (ablative first-person possessive singular)
qüvvətlərimdən (ablative first-person possessive singular)
qüvvətindən (ablative possessive second-person singular)
qüvvətlərindən (ablative possessive second-person singular)
qüvvətindən (ablative possessive singular third-person)
qüvvətlərindən (ablative possessive singular third-person)
qüvvətimizdən (ablative first-person plural possessive)
qüvvətlərimizdən (ablative first-person plural possessive)
qüvvətinizdən (ablative plural possessive second-person)
qüvvətlərinizdən (ablative plural possessive second-person)
qüvvətindən (ablative plural possessive third-person)
qüvvətlərindən (ablative plural possessive third-person)
qüvvətimin (first-person genitive possessive singular)
qüvvətlərimin (first-person genitive possessive singular)
qüvvətinin (genitive possessive second-person singular)
qüvvətlərinin (genitive possessive second-person singular)
qüvvətinin (genitive possessive singular third-person)
qüvvətlərinin (genitive possessive singular third-person)
qüvvətimizin (first-person genitive plural possessive)
qüvvətlərimizin (first-person genitive plural possessive)
qüvvətinizin (genitive plural possessive second-person)
qüvvətlərinizin (genitive plural possessive second-person)
qüvvətinin (genitive plural possessive third-person)
qüvvətlərinin (genitive plural possessive third-person)

From Arabic قُوَّة (quwwa). Doublet of qüvvə.

(algebra) degree (the sum of the exponents of a term)


underlying set

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


घनीय

Sanskritic formation from घन (ghan, “cube”) + -ईय (-īya).

sa-af

घनीय बहुपद


formal power series

en-noun

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.


łączny

pl-adj

łączny (not comparable, derived adverb łącznie)

łączny (masculine nominative singular)
łączna (feminine nominative singular)
łączne (neuter nominative singular)
łączni (nominative plural virile)
łączne (nominative plural)
łącznego (genitive masculine singular)
łącznej (feminine genitive singular)
łącznego (genitive neuter singular)
łącznych (genitive plural)
łącznych (genitive plural virile)
łącznemu (dative masculine singular)
łącznej (dative feminine singular)
łącznemu (dative neuter singular)
łącznym (dative plural)
łącznym (dative plural virile)
łącznego (accusative animate masculine singular)
łączny (accusative inanimate masculine singular)
łączną (accusative feminine singular)
łączne (accusative neuter singular)
łącznych (accusative plural virile)
łączne (accusative plural)
łącznym (instrumental masculine singular)
łączną (feminine instrumental singular)
łącznym (instrumental neuter singular)
łącznymi (instrumental plural)
łącznymi (instrumental plural virile)
łącznym (locative masculine singular)
łącznej (feminine locative singular)
łącznym (locative neuter singular)
łącznych (locative plural)
łącznych (locative plural virile)


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

en-noun

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.


bậc

北 (CJK)
堛 (CJK)
𨄑 (CJK)
弼 (CJK)
bực (alternative)

bậc 1

bậc 2

bậc 3

(algebra, of a monomial, polynomial, equality, equation, function, etc.) a degree; an order

polynomial; equality; equation; function; polynomial; equality; equation; function


lokalisasyon

multiplicative inverse
multiplicative inverse

(algebra) a systematic method of adding multiplicative inverses to a ring


diskriminant

diskriminant (nominative singular)
diskriminanty (nominative plural)
diskriminantu (genitive singular)
diskriminantů (genitive plural)
diskriminantu (dative singular)
diskriminantům (dative plural)
diskriminant (accusative singular)
diskriminanty (accusative plural)
diskriminante (singular vocative)
diskriminanty (plural vocative)
diskriminantu (locative singular)
diskriminantech (locative plural)
diskriminantem (instrumental singular)
diskriminanty (instrumental plural)

(algebra) discriminant (expression that gives information about the roots of a polynomial)


diskriminant

head

diskriminant c (singular definite diskriminanten, plural indefinite diskriminanter)

diskriminanten (definite singular)
diskriminanter (indefinite plural)
diskriminant (indefinite nominative singular)
diskriminanten (definite nominative singular)
diskriminanter (indefinite nominative plural)
diskriminanterne (definite nominative plural)
diskriminants (genitive indefinite singular)
diskriminantens (definite genitive singular)
diskriminanters (genitive indefinite plural)
diskriminanternes (definite genitive plural)


代數結構

ㄉㄞˋ ㄕㄨˋ ㄐㄧㄝˊ ㄍㄡˋ

ㄉㄞˋ ㄕㄨˋ ㄐㄧㄝˊ ㄍㄡˋ

дайшу цзегоу

/taɪ̯⁵¹⁻⁵³ ʂu⁵¹ t͡ɕi̯ɛ³⁵ koʊ̯⁵¹/

/tɔːi̯²² sou̯³³ kiːt̚³ kʰɐu̯³³/

/tɔːi̯²² sou̯³³ kiːt̚³ kɐu̯³³/

/taɪ̯⁵¹⁻⁵³ ʂu⁵¹ t͡ɕi̯ɛ³⁵ koʊ̯⁵¹/

/tɔːi̯²² sou̯³³ kiːt̚³ kʰɐu̯³³/

/tɔːi̯²² sou̯³³ kiːt̚³ kɐu̯³³/


free Boolean algebra

en-noun

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.


нормирование

ru-noun+

нормирова́ние

норми́рование

нормирова́ние or норми́рование • (normirovánije or normírovanije) n inan (genitive нормирова́ния or норми́рования, nominative plural нормирова́ния or норми́рования, genitive plural нормирова́ний or норми́рований)

нормирова́ние (canonical)
норми́рование (canonical)
normirovánije (romanization)
normírovanije (romanization)
нормирова́ния (genitive)
норми́рования (genitive)
нормирова́ния (nominative plural)
норми́рования (nominative plural)
нормирова́ний (genitive plural)
норми́рований (genitive plural)
нормирова́ние (nominative singular)
норми́рование (nominative singular)
нормирова́ния (nominative plural)
норми́рования (nominative plural)
нормирова́ния (genitive singular)
норми́рования (genitive singular)
нормирова́ний (genitive plural)
норми́рований (genitive plural)
нормирова́нию (dative singular)
норми́рованию (dative singular)
нормирова́ниям (dative plural)
норми́рованиям (dative plural)
нормирова́ние (accusative singular)
норми́рование (accusative singular)
нормирова́ния (accusative plural)
норми́рования (accusative plural)
нормирова́нием (instrumental singular)
норми́рованием (instrumental singular)
нормирова́ниями (instrumental plural)
норми́рованиями (instrumental plural)
нормирова́нии (prepositional singular)
норми́ровании (prepositional singular)
нормирова́ниях (plural prepositional)
норми́рованиях (plural prepositional)

From нормирова́ть (normirovátʹ) + -а́ние (-ánije) (suffix-stressed) or норми́ровать (normírovatʹ) + -ание (-anije) (stem-stressed).

af

нормирова́ть

норми́ровать

(algebra) absolute value (type of function in advanced algebra, a generalization of the basic concept of absolute value)


free abelian group

en-noun

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


mettre en évidence

LL-Q150 (fra)-Lyokoï-mettre en évidence.wav


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.


降冪

ja-noun

降(こう)冪(べき) • (kōbeki) ^(←かうべき (kaubeki)?)

降冪 (canonical)
kōbeki (romanization)
kaubeki (romanization)
かうべき (hiragana historical)


conjugacy class

en-noun

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.


進位

/t͡ɕin⁵¹⁻⁵³ weɪ̯⁵¹/

/t͡sɵn³³ wɐi̯²²⁻³⁵/

/t͡sɵn³³ wɐi̯²²/

/t͡ɕin⁵¹⁻⁵³ weɪ̯⁵¹/

/t͡sɵn³³ wɐi̯²²⁻³⁵/

/t͡sɵn³³ wɐi̯²²/

(algebra) to carry (number during addition, etc.); to carry over


đa thức

[ʔɗaː˧˧ tʰɨk̚˧˦]

[ʔɗaː˧˧ tʰɨk̚˦˧˥]

[ʔɗaː˧˧ tʰɨk̚˦˥]

LL-Q9199 (vie)-Penn Zero MSSJ-đa thức.wav


đơn thức

head

đơn thức • (單式)

vi-noun

đơn thức • (單式)

[ʔɗəːn˧˧ tʰɨk̚˧˦]

[ʔɗəːŋ˧˧ tʰɨk̚˦˧˥]

[ʔɗəːŋ˧˧ tʰɨk̚˦˥]

LL-Q9199 (vie)-Penn Zero MSSJ-đơn thức.wav


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.


radical ideal

1981, Harry C. Hutchins, Examples of Commutative Rings, Polygonal Publishing House, page 3,
Any intersection of radical ideals is again a radical ideal. Since not every intersection of prime ideals is a prime ideal, it follows that not all radical ideals are prime.

It follows at once that R#47;I is a reduced ring iff I is a radical ideal. Thus, the ideals I(X) are all radical ideals.

1995, David Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Springer, page 33:

1997, D. D. Anderson, Dong Je Kwak, Some Remarks on G-Noetherian Rings, Paul-Jean Cahen, Marco Fontana, Evan Houston, Salah-Eddine Kabbaj (editors), Commutative Ring Theory: Proceedings of the II International Conference, Marcel Dekker, page 30,
Thus Theorem 2 [8] is a special case of the well-known result that a commutative ring R satisfies the ascending chain condition on radical ideals if and only if R satisfies the ascending chain condition on prime ideals and each radical ideal of R is a finite intersection of prime ideals […] .

(algebra, commutative algebra, ring theory) An ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rⁿ ∈ I for some positive integer n, then r ∈ I).

commutative algebra; ring theory; commutative algebra; ring theory


reduced ring

We do this for reduced rings in Corollary 10, and for rings with nonzero nilpotents in Corollary 15.

1997, Thomas G. Lucas, “Characterizing When R(X) is Completely Integrally Closed”, in Daniel Anderson, editor, Factorization in Integral Domains, Marcel Dekker, page 401:

Extending the notion of a reduced ring, we call a right module M over a ring R a reduced module if, for any m#92;inM and a#92;inR, ma#61;0 implies mR#92;capMa#61;0. Various results of reduced rings are extended to reduced modules.

2004, Tsiu-Kwen Lee, Yiqiang Zhou, “Reduced Modules”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, Marcel Dekker, page 365:

In general, the first case of importance is the normalization of a reduced ring R in its quotient ring K(R).

2005, David Eisenbud, The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry, Springer, page 210:

(algebra, ring theory) A ring R that has no nonzero nilpotent elements; equivalently, such that, for x ∈ R, x² = 0 implies x = 0.

reduzierter Ring

ring that has no nonzero nilpotent elements


reduced algebra

en-noun

reduced algebra (plural reduced algebras)

(algebra, ring theory) An algebra over a reduced ring.


交換環

ㄐㄧㄠ ㄏㄨㄢˋ ㄏㄨㄢˊ

ㄐㄧㄠ ㄏㄨㄢˋ ㄏㄨㄢˊ

цзяохуаньхуань

/t͡ɕi̯ɑʊ̯⁵⁵ xu̯än⁵¹ xu̯än³⁵/

/t͡ɕi̯ɑʊ̯⁵⁵ xu̯än⁵¹ xu̯än³⁵/


整環

/ʈ͡ʂɤŋ²¹⁴⁻²¹ xu̯än³⁵/

/t͡sɪŋ³⁵ waːn²¹/

/ʈ͡ʂɤŋ²¹⁴⁻²¹ xu̯än³⁵/

/t͡sɪŋ³⁵ waːn²¹/


阿貝爾群

ㄚ ㄅㄟˋ ㄦˇ ㄑㄩㄣˊ

ㄚ ㄅㄟˋ ㄦˇ ㄑㄩㄣˊ

/ˀä⁵⁵ peɪ̯⁵¹ ˀɤɻ²¹⁴⁻²¹ t͡ɕʰyn³⁵/

/ˀä⁵⁵ peɪ̯⁵¹ ˀɤɻ²¹⁴⁻²¹ t͡ɕʰyn³⁵/


陪集

/pʰeɪ̯³⁵ t͡ɕi³⁵/

/pʰeɪ̯³⁵ t͡ɕi³⁵/


bounded lattice

bounded lattice (plural bounded lattices)

2004, Anna Maria Radzikowska, Etienne Kerre, On L-Fuzzy Rough Sets, Leszek Rutkowski, Jörg Siekmann, Ryszard Tadeusiewicz, Lotfi A. Zadeh (editors), Artificial Intelligence and Soft Computing — ICAISC 2004: 7th International Conference, Proceedings, Springer, LNAI 3070, page 526,
A residuated lattice is an extension of a bounded lattice by a monoid operation and its residuum, which are abstract counterparts of a triangular norm and a fuzzy residual implication, respectively.

2006, Bart Van Gasse, Chris Cornelis, Glad Deschrijver, Etienne Kerre, Triangle Lattices: Towards an Axiomatization of Interval-Valued Residuated Lattices, Salavatore Greco, Yukata Hata, Shoji Hirano, Masahiro Inuiguchi, Sadaaki Miyamoto, Hung Son Nguyen, Roman Słowiński (editors), Rough Sets and Current Trends in Computing: 5th International Conference, Proceedings, Springer, LNAI 4259, page 117,
Indeed, in the scope of these logics, formulas can be assigned not only 0 and 1 as truth values, but also elements of [0,1], or, more generally, of a bounded lattice ℒ.

In this paper, we study idempotent nullnorms on bounded lattices. We prove that there is no idempotent nullnorm on a distributive bounded lattice L different from the proposal in [13].

2018, Gül Deniz Çayli, Funda Karaçal, “Some Remarks on Idempotent Nullnorms on Bounded Lattices”, in Vicenç Torra, Radko Mesiar, Bernard De Baets, editors, Aggregation Functions in Theory and in Practice, Springer,, page 32:

partially ordered set
partially ordered set

(algebra, order theory) Any lattice (type of partially ordered set) that has both a greatest and a least element.


nhị thức

[ɲi˧˨ʔ tʰɨk̚˧˦]

[ɲɪj˨˩ʔ tʰɨk̚˦˧˥]

[ɲɪj˨˩˨ tʰɨk̚˦˥]

LL-Q9199 (vie)-Penn Zero MSSJ-nhị thức.wav

nhị thức Niu-tơn


tam thức

[taːm˧˧ tʰɨk̚˧˦]

[taːm˧˧ tʰɨk̚˦˧˥]

[taːm˧˧ tʰɨk̚˦˥]

LL-Q9199 (vie)-Penn Zero MSSJ-tam thức.wav


hằng đẳng thức

[haŋ˨˩ ʔɗaŋ˧˩ tʰɨk̚˧˦]

[haŋ˦˩ ʔɗaŋ˧˨ tʰɨk̚˦˧˥]

[haŋ˨˩ ʔɗaŋ˨˩˦ tʰɨk̚˦˥]

LL-Q9199 (vie)-Penn Zero MSSJ-hằng đẳng thức.wav

Sino-Vietnamese word from 恆等式, composed of 恆 (“fixed”) and 等式 (“equality”)

vi-etym-sino

hằng đẳng thức đáng nhớ

vi:Bảy hằng đẳng thức đáng nhớ
vi:hằng đẳng thức


field extension

en-noun

field extension (plural field extensions)

A Galois field extension may be defined in terms of its Galois group (Section 2) or in terms of the internal structure of the extension (Section 3).

1974, Thomas W. Hungerford, Algebra, Springer, page 230:

1998, David Goss, Basic Structures of Function Field Arithmetic, Springer, Corrected 2nd Printing, page 283,
Note that the extension of L obtained by adjoining all division points of ψ includes at most a finite constant field extension.

2007, Pierre Antoine Grillet, Abstract Algebra, Springer, 2bd Edition, page 530,
A field extension of a field K is, in particular, a K-algebra. Hence any two field extensions of K have a tensor product that is a K-algebra.

(algebra, field theory, algebraic geometry) Any pair of fields, denoted L/K, such that K is a subfield of L.

kuntalaajennus

pair of fields such that one is a subfield of the other


prime field

1968, J. F. Schuh, Mathematical Tools for Modern Physics, Philips Technical Library, page 91,
Theorem 2. Each field contains a unique prime field.

2004, H. Gopalkrishna Gadiyar, K. M. Sangeeta Maini, R. Padma, Cryptography, Connections, Cocycles and Crystals: A p-adic Exploration of the Discrete Logarithm Problem, Anne Canteaut, Kapaleeswaran Viswanathan (editors), Progress in Cryptology - INDOCRYPT 2004: 5th International Conference, Proceedings, Springer, LNCS 3348, page 305,
Applying Hensel's lemma to the discrete logarithm problem over prime fields reveals the rich geometric and algebraic structure underlying the problem.

Since every field contains a prime field, by studying prime fields, and the way properties are passed over to extensions, Steinitz would attain a full picture of the structure of all possible fields.

2004, Leo Corry, Modern Algebra and the Rise of Mathematical Structures, 2nd edition, Springer (Birkhäuser), page 194:

(algebra, field theory) A field that contains no proper subfields.


algebraic extension

algebraic extension (plural algebraic extensions)

What we now have to prove is that: if an overring R#42; of R is such that R#42; is an integral domain, R#42; is integral over R and the quotient field #92;mathfrak#123;L#125;#42; of R#42; is a finite algebraic extension of #92;mathfrak#123;L#125;, then R#42; is a local ring and a finite R-module.

1964, Shreeram Shankar Abhyankar, Local Analytic Geometry, Academic Press, page 200:

It turns out that when an irreducible polynomial splits in an algebraic extension, then the order at the new irreducible factors remains the same as before for arguments that are defined over the ground field.

2005, Manuel Bronstein, Symbolic Integration I: Transcendental Functions, 2nd edition, Springer, page 110:

We describe the correspondence between subfields of the field P#95;a(O) that are algebraic extensions of the field K(X) and the subgroups of finite index of the fundamental group #92;pi#95;1(X#92;setminusO).

2013, Askold Khovanskii, Galois Theory, Coverings, and Riemann Surfaces, Springer, page 65:

(algebra, field theory) A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K).


discrete valuation

discrete valuation (plural discrete valuations)

Therefore the theory branches in two essential parts : first the study of discrete valuations of the Weyl field D#95;1(K) extending a discrete valuation of K, secondly the discrete valuations of D#95;1(K) having the purely transcendental field K(t) as a residue field.

2000, Freddy Van Oystaeyen, Algebraic Geometry for Associative Algebras, CRC Press, page 173:

One of the few possibilities to extend some arithmetic aspects of number theory to more abstract fields is via valuation theory and, as a first step, we may restrict attention to discrete valuations.

2004, Farahat S. Aly, Freddy Van Oystaeyen, “Hopf Order Module Algebra Orders”, in Stefaan Caenepeel, Fred Van Oystaeyen, editors, Hopf Algebras in Noncommutative Geometry and Physics, CRC Press, page 21:

Let k be a field with a nontrivial discrete valuation which is complete and has perfect residue field.

2011, Ricardo Portilla, “Parametrizing nilpotent orbits in p-adic symmetric spaces”, in Robert S. Doran, Paul J. Sally, Jr., Loren Spice, editors, Harmonic Analysis on Reductive, p-adic Groups, American Mathematical Society, page 201:

(algebra, field theory) A valuation (on some field) that takes integer values (including infinity).


combinatorial commutative algebra

combinatorial commutative algebra (uncountable)

This construction has become a fundamental tool in combinatorial commutative algebra due to the work of Stanley, Hochster and Reisner ([Ho77], [S75], [S96], [R76]).

2011, Viviana Ene, Jũrgen Herzog, Gröbner Bases in Commutative Algebra, American Mathematical Society, page 88:

2013, Anna M. Bigatti, Philippe Gimenez, Eduardo Sáenz-de-Cabezón Introduction, Anna M. Bigatti, Philippe Gimenez, Eduardo Sáenz-de-Cabezón (editors), Monomial Ideals, Computations and Applications, Springer, page vii,
Monomial ideals and algebras are among the simplest structures in commutative algebra and the main objects in combinatorial commutative algebra.

Using techniques from combinatorial commutative algebra [MS05, KM], they showed that certain lattice games admit a particular type of algebraic decomposition known as affine stratifications, from which a finitely representable winning strategy can be recovered.

2013, Aaron N. Siegel, Combinatorial Game Theory, American Mathematical Society, page 269:

(algebra) A relatively new discipline in mathematics that combines techniques and concepts from combinatorics and commutative algebra, and in which the geometry of convex polytopes also plays a significant role.


zero divisor

zero divisors (plural)
zero-divisor (alternative)

regular element

antonym(s) of “any element whose product with some nonzero element is zero”

trivial zero divisor

An idempotent element e#92;ne 1 of a ring is always a (two-sided) zero divisor, since e(1-e)#61;0#61;(1-e)e.

Linnell [25, 1977] proved that if G is a torsion-free abelian by locally finite by super-solvable group and K is any field, then K[G] has no nontrivial zero divisors.

1984, J. B. Srivastava, “23: Projective Modules, Zero Divisors, and Noetherian Group Algebras”, in Dinesh N. Manocha, editor, Algebra and its Applications, CRC Press, page 170:

In the ring of integers, there are no zero divisors except 0. In a ring obtained from a Boolean algebra, on the other hand, every element except the identity is a zero-divisor.
The concept of a zero-divisor is intimately related to cancellation law as we see n the following proposition.
1.7 Proposition: Let R be a ring and x#92;inR. Then for all y,x#92;inR, either of the equations xy#61;xz or yx#61;zx implies y#61;z if and only if x is not a zero divisor. In other words, cancellation by an element is possible iff it is not a zero-divisor.

1989, K. D. Joshi, Foundations of Discrete Mathematics, New Age International, page 390:

In [1], Anderson and Livingston introduced and studied the zero-divisor graph whose vertices are the non-zero zero-divisors.

2010, Mitsuo Kanemitsu, “The Number of Distinct 4-Cycles and 2-Matchings of Some Zero Divisor Graphs”, in Masami Ito, Yuji Kobayashi, Kunitaka Shoji, editors, Automata, Formal Languages and Algebraic Systems: Proceedings of AFLAS 2008, World Scientific, page 63:

(algebra, ring theory) An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

If R is an integral domain, that is, has no zero divisors, then R#91;x#93; also has no zero divisors.

2000, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 2nd edition, Springer, page 234:

An element a of a ring is called a zero-divisor if a#92;ne 0 and ab#61;0 or ba#61;0 for some b#92;ne 0; if a is neither 0 nor a zero-divisor, it is said to be regular (see Section 7.1). A non-trivial ring without zero-divisors is called an integral domain; this term is not taken to imply commutativity.

2002, Paul M. Cohn, Further Algebra and Applications, Springer, page xi:

If a and b are non-zero elements of R such that ab#61;0, then aandb are both called zero divisors. If R is non-trivial and has no zero divisors, then it is called an integral domain. Note that if a is a unit in R, it cannot be a zero divisor (if ab#61;0, then multiplying both sides of this equation by a#123;-1#125; yields b#61;0.

2009, Victor Shoup, A Computational Introduction to Number Theory and Algebra, 2nd edition, Cambridge University Press, page 171:

(algebra, ring theory) A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.


mise en évidence

fr-noun

mise en évidence f (plural mises en évidence)


ordered ring

1965, Seth Warner, Modern Algebra, Dover, 1990, Single-volume republication, page 217,
If < is an ordering on A compatible with its ring structure, we shall say that (A,+,·,<) is an ordered ring. An element x of an ordered ring A is positive if x>0, and x is strictly positive if x>0.
The set of all positive elements of an ordered ring A is denoted by A_+, and the set of all strictly positive elements of A is denoted by A^*₊.
If (A,+,·,<) is an ordered ring and if < is a total ordering, we shall, of course, call (A,+,·,<) a totally ordered ring; if (A,+,·) is a field, we shall call (A,+,·,<) an ordered field, and if, moreover, < is a total ordering, we shal call (A,+·,<) a totally ordered field.

(OR) The relations x>0 and y>0 imply xy>0.
The ring A, together with such an ordering, is called an ordered ring.
Examples. — 1) The rings Q and Z , with the usual orderings, are ordered rings.
2) A product of ordered rings, equipped with the product ordering, is an ordered ring. In particular, the ring Aᴱ of mappings from a set E to an ordered ring A is an ordered ring.
3) A subring of an ordered ring, with the induced ordering, is an ordered ring.

1990, P. M. Cohn, J. Howie (translators), Nicolas Bourbaki, Algebra II: Chapters 4-7, [1981, N. Bourbaki, Algèbre, Chapitres 4 à 7, Masson], Springer, 2003, Softcover reprint, page 19, DEFINITION 1. — Given a commutative ring A, we say that an ordering on A is compatible with the ring structure on A if it is compatible with the additive group structure of A, and if it satisfies the following axiom

(algebra, order theory, ring theory) A ring, R, equipped with a partial order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.

The positive elements in an ordered ring allow us to compare elements to 0, but we know in the integers that we can compare any two elements to each other. For example, we know that 4gt;2 because 4-2gt;0. We can extend this idea to any ordered ring. If R is an ordered ring and a,b#92;inR, then we know by trichotomy that exactly one of the following must be true: a-bgt;0, a-b#61;0, or -(a-b)gt;0.

2013, Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom, Abstract Algebra: An Inquiry Based Approach, CRC Press, page 253:

(1) The set R⁺ is closed under addition and multiplication.
(2) If x∈R then exactly one of the following is true: (trichotomy law)
(a) x=0,
(b) x∈R⁺,
(c) -x∈R⁺.
If further R is an integral domain we call R an ordered integral domain.
[…]
Lemma 3.5.9. If R is an ordered ring and a∈R is a positive element, then the set na:n∈ N⊂R⁺.
[…]
Theorem 3.5.2. An ordered ring must be infinite.

2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra, Johns Hopkins University Press, page 77, Definition 3.5.4. A ring R is an ordered ring if there exists a distinguished set R⁺, R⁺⊂R, called the set of positive elements, with the properties that

(algebra, order theory, ring theory) A ring, R, equipped with a total order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.


invariant theory

en-noun

invariant theory (countable and uncountable, plural invariant theories)

1993, Bernd Sturmfels, Introduction, David Hilbert, Reinhard C. Laubenbacher (translator and editor), Bernd Sturmfels (editor), Theory of Algebraic Invariants, Cambridge University Press, page xi,
Today, invariant theory is often understood as a branch of representation theory, algebraic geometry, commutative algebra, and algebraic combinatorics. Each of these four disciplines has roots in nineteenth-century invariant theory. […] In modern terms, the basic problem of invariant theory can be categorized as follows. Let V be a K-vector space on which a group G acts linearly. In the ring of polynomial functions K[V] consider the subring K[V]ᴳ consisting of all polynomial functions on V which are invariant under the action of the group G. The basic problem is to describe the invariant ring K[V]ᴳ. In particular, we would like to know whether K[V]ᴳ is finitely generated as a K-algebra and, if so, to give an algorithm for computing generators.

Invariant theory is the great romantic story of mathematics.[…]In our century, Lie theory and algebraic geometry, differential algebra and algebraic combinatorics are all offsprings of invariant theory.

2001, Gian-Carlo Rota, “What is invariant theory, really?”, in H. Crapo, D. Senato, editors, Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota, Springer,, page 41:

For a linear algebraic group G and a regular representation (#92;rho,V) of G, the basic problem of invariant theory is to describe the G-invariant elements (#123;#92;bigotimes#125;ᵏⱽ)ᴳ of the k-fold tensor product for all k.

2009, Roe Goodman, Nolan R. Wallach, Symmetry, Representations, and Invariants, Springer, page 225:

(algebra, representation theory) The branch of algebra concerned with actions of groups on algebraic varieties from the point of view of their effect on functions.


kokonaisalue

kokonaisalue (nominative singular)
kokonaisalueet (nominative plural)
kokonaisalue (accusative nominative singular)
kokonaisalueet (accusative nominative plural)
kokonaisalueen (accusative genitive singular)
kokonaisalueet (accusative genitive plural)
kokonaisalueen (genitive singular)
kokonaisalueiden (genitive plural)
kokonaisalueitten (genitive plural)
kokonaisaluetta (partitive singular)
kokonaisalueita (partitive plural)
kokonaisalueessa (inessive singular)
kokonaisalueissa (inessive plural)
kokonaisalueesta (elative singular)
kokonaisalueista (elative plural)
kokonaisalueeseen (illative singular)
kokonaisalueisiin (illative plural)
kokonaisalueihin (illative plural)
kokonaisalueella (adessive singular)
kokonaisalueilla (adessive plural)
kokonaisalueelta (ablative singular)
kokonaisalueilta (ablative plural)
kokonaisalueelle (allative singular)
kokonaisalueille (allative plural)
kokonaisalueena (essive singular)
kokonaisalueina (essive plural)
kokonaisalueeksi (singular translative)
kokonaisalueiksi (plural translative)
kokonaisalueetta (abessive singular)
kokonaisalueitta (abessive plural)
kokonaisaluein (instructive plural)
See the possessive forms below. (comitative)
kokonaisalueeni (nominative singular)
kokonaisalueeni (nominative plural)
kokonaisalueeni (accusative nominative singular)
kokonaisalueeni (accusative nominative plural)
kokonaisalueeni (accusative genitive singular)
kokonaisalueeni (accusative genitive plural)
kokonaisalueeni (genitive singular)
kokonaisalueideni (genitive plural)
kokonaisalueitteni (genitive plural)
kokonaisaluettani (partitive singular)
kokonaisalueitani (partitive plural)
kokonaisalueessani (inessive singular)
kokonaisalueissani (inessive plural)
kokonaisalueestani (elative singular)
kokonaisalueistani (elative plural)
kokonaisalueeseeni (illative singular)
kokonaisalueisiini (illative plural)
kokonaisalueihini (illative plural)
kokonaisalueellani (adessive singular)
kokonaisalueillani (adessive plural)
kokonaisalueeltani (ablative singular)
kokonaisalueiltani (ablative plural)
kokonaisalueelleni (allative singular)
kokonaisalueilleni (allative plural)
kokonaisalueenani (essive singular)
kokonaisalueinani (essive plural)
kokonaisalueekseni (singular translative)
kokonaisalueikseni (plural translative)
kokonaisalueettani (abessive singular)
kokonaisalueittani (abessive plural)
kokonaisalueineni (comitative plural)
kokonaisalueesi (nominative singular)
kokonaisalueesi (nominative plural)
kokonaisalueesi (accusative nominative singular)
kokonaisalueesi (accusative nominative plural)
kokonaisalueesi (accusative genitive singular)
kokonaisalueesi (accusative genitive plural)
kokonaisalueesi (genitive singular)
kokonaisalueidesi (genitive plural)
kokonaisalueittesi (genitive plural)
kokonaisaluettasi (partitive singular)
kokonaisalueitasi (partitive plural)
kokonaisalueessasi (inessive singular)
kokonaisalueissasi (inessive plural)
kokonaisalueestasi (elative singular)
kokonaisalueistasi (elative plural)
kokonaisalueeseesi (illative singular)
kokonaisalueisiisi (illative plural)
kokonaisalueihisi (illative plural)
kokonaisalueellasi (adessive singular)
kokonaisalueillasi (adessive plural)
kokonaisalueeltasi (ablative singular)
kokonaisalueiltasi (ablative plural)
kokonaisalueellesi (allative singular)
kokonaisalueillesi (allative plural)
kokonaisalueenasi (essive singular)
kokonaisalueinasi (essive plural)
kokonaisalueeksesi (singular translative)
kokonaisalueiksesi (plural translative)
kokonaisalueettasi (abessive singular)
kokonaisalueittasi (abessive plural)
kokonaisalueinesi (comitative plural)
kokonaisalueemme (nominative singular)
kokonaisalueemme (nominative plural)
kokonaisalueemme (accusative nominative singular)
kokonaisalueemme (accusative nominative plural)
kokonaisalueemme (accusative genitive singular)
kokonaisalueemme (accusative genitive plural)
kokonaisalueemme (genitive singular)
kokonaisalueidemme (genitive plural)
kokonaisalueittemme (genitive plural)
kokonaisaluettamme (partitive singular)
kokonaisalueitamme (partitive plural)
kokonaisalueessamme (inessive singular)
kokonaisalueissamme (inessive plural)
kokonaisalueestamme (elative singular)
kokonaisalueistamme (elative plural)
kokonaisalueeseemme (illative singular)
kokonaisalueisiimme (illative plural)
kokonaisalueihimme (illative plural)
kokonaisalueellamme (adessive singular)
kokonaisalueillamme (adessive plural)
kokonaisalueeltamme (ablative singular)
kokonaisalueiltamme (ablative plural)
kokonaisalueellemme (allative singular)
kokonaisalueillemme (allative plural)
kokonaisalueenamme (essive singular)
kokonaisalueinamme (essive plural)
kokonaisalueeksemme (singular translative)
kokonaisalueiksemme (plural translative)
kokonaisalueettamme (abessive singular)
kokonaisalueittamme (abessive plural)
kokonaisalueinemme (comitative plural)
kokonaisalueenne (nominative singular)
kokonaisalueenne (nominative plural)
kokonaisalueenne (accusative nominative singular)
kokonaisalueenne (accusative nominative plural)
kokonaisalueenne (accusative genitive singular)
kokonaisalueenne (accusative genitive plural)
kokonaisalueenne (genitive singular)
kokonaisalueidenne (genitive plural)
kokonaisalueittenne (genitive plural)
kokonaisaluettanne (partitive singular)
kokonaisalueitanne (partitive plural)
kokonaisalueessanne (inessive singular)
kokonaisalueissanne (inessive plural)
kokonaisalueestanne (elative singular)
kokonaisalueistanne (elative plural)
kokonaisalueeseenne (illative singular)
kokonaisalueisiinne (illative plural)
kokonaisalueihinne (illative plural)
kokonaisalueellanne (adessive singular)
kokonaisalueillanne (adessive plural)
kokonaisalueeltanne (ablative singular)
kokonaisalueiltanne (ablative plural)
kokonaisalueellenne (allative singular)
kokonaisalueillenne (allative plural)
kokonaisalueenanne (essive singular)
kokonaisalueinanne (essive plural)
kokonaisalueeksenne (singular translative)
kokonaisalueiksenne (plural translative)
kokonaisalueettanne (abessive singular)
kokonaisalueittanne (abessive plural)
kokonaisalueinenne (comitative plural)
kokonaisalueensa (nominative singular)
kokonaisalueensa (nominative plural)
kokonaisalueensa (accusative nominative singular)
kokonaisalueensa (accusative nominative plural)
kokonaisalueensa (accusative genitive singular)
kokonaisalueensa (accusative genitive plural)
kokonaisalueensa (genitive singular)
kokonaisalueidensa (genitive plural)
kokonaisalueittensa (genitive plural)
kokonaisaluettaan (partitive singular)
kokonaisaluettansa (partitive singular)
kokonaisalueitaan (partitive plural)
kokonaisalueitansa (partitive plural)
kokonaisalueessaan (inessive singular)
kokonaisalueessansa (inessive singular)
kokonaisalueissaan (inessive plural)
kokonaisalueissansa (inessive plural)
kokonaisalueestaan (elative singular)
kokonaisalueestansa (elative singular)
kokonaisalueistaan (elative plural)
kokonaisalueistansa (elative plural)
kokonaisalueeseensa (illative singular)
kokonaisalueisiinsa (illative plural)
kokonaisalueihinsa (illative plural)
kokonaisalueellaan (adessive singular)
kokonaisalueellansa (adessive singular)
kokonaisalueillaan (adessive plural)
kokonaisalueillansa (adessive plural)
kokonaisalueeltaan (ablative singular)
kokonaisalueeltansa (ablative singular)
kokonaisalueiltaan (ablative plural)
kokonaisalueiltansa (ablative plural)
kokonaisalueelleen (allative singular)
kokonaisalueellensa (allative singular)
kokonaisalueilleen (allative plural)
kokonaisalueillensa (allative plural)
kokonaisalueenaan (essive singular)
kokonaisalueenansa (essive singular)
kokonaisalueinaan (essive plural)
kokonaisalueinansa (essive plural)
kokonaisalueekseen (singular translative)
kokonaisalueeksensa (singular translative)
kokonaisalueikseen (plural translative)
kokonaisalueiksensa (plural translative)
kokonaisalueettaan (abessive singular)
kokonaisalueettansa (abessive singular)
kokonaisalueittaan (abessive plural)
kokonaisalueittansa (abessive plural)
kokonaisalueineen (comitative plural)
kokonaisalueinensa (comitative plural)


kvasiryhmä

kvasiryhmä (nominative singular)
kvasiryhmät (nominative plural)
kvasiryhmä (accusative nominative singular)
kvasiryhmät (accusative nominative plural)
kvasiryhmän (accusative genitive singular)
kvasiryhmät (accusative genitive plural)
kvasiryhmän (genitive singular)
kvasiryhmien (genitive plural)
kvasiryhmäin (genitive plural rare)
kvasiryhmää (partitive singular)
kvasiryhmiä (partitive plural)
kvasiryhmässä (inessive singular)
kvasiryhmissä (inessive plural)
kvasiryhmästä (elative singular)
kvasiryhmistä (elative plural)
kvasiryhmään (illative singular)
kvasiryhmiin (illative plural)
kvasiryhmällä (adessive singular)
kvasiryhmillä (adessive plural)
kvasiryhmältä (ablative singular)
kvasiryhmiltä (ablative plural)
kvasiryhmälle (allative singular)
kvasiryhmille (allative plural)
kvasiryhmänä (essive singular)
kvasiryhminä (essive plural)
kvasiryhmäksi (singular translative)
kvasiryhmiksi (plural translative)
kvasiryhmättä (abessive singular)
kvasiryhmittä (abessive plural)
kvasiryhmin (instructive plural)
See the possessive forms below. (comitative)
kvasiryhmäni (nominative singular)
kvasiryhmäni (nominative plural)
kvasiryhmäni (accusative nominative singular)
kvasiryhmäni (accusative nominative plural)
kvasiryhmäni (accusative genitive singular)
kvasiryhmäni (accusative genitive plural)
kvasiryhmäni (genitive singular)
kvasiryhmieni (genitive plural)
kvasiryhmäini (genitive plural rare)
kvasiryhmääni (partitive singular)
kvasiryhmiäni (partitive plural)
kvasiryhmässäni (inessive singular)
kvasiryhmissäni (inessive plural)
kvasiryhmästäni (elative singular)
kvasiryhmistäni (elative plural)
kvasiryhmääni (illative singular)
kvasiryhmiini (illative plural)
kvasiryhmälläni (adessive singular)
kvasiryhmilläni (adessive plural)
kvasiryhmältäni (ablative singular)
kvasiryhmiltäni (ablative plural)
kvasiryhmälleni (allative singular)
kvasiryhmilleni (allative plural)
kvasiryhmänäni (essive singular)
kvasiryhminäni (essive plural)
kvasiryhmäkseni (singular translative)
kvasiryhmikseni (plural translative)
kvasiryhmättäni (abessive singular)
kvasiryhmittäni (abessive plural)
kvasiryhmineni (comitative plural)
kvasiryhmäsi (nominative singular)
kvasiryhmäsi (nominative plural)
kvasiryhmäsi (accusative nominative singular)
kvasiryhmäsi (accusative nominative plural)
kvasiryhmäsi (accusative genitive singular)
kvasiryhmäsi (accusative genitive plural)
kvasiryhmäsi (genitive singular)
kvasiryhmiesi (genitive plural)
kvasiryhmäisi (genitive plural rare)
kvasiryhmääsi (partitive singular)
kvasiryhmiäsi (partitive plural)
kvasiryhmässäsi (inessive singular)
kvasiryhmissäsi (inessive plural)
kvasiryhmästäsi (elative singular)
kvasiryhmistäsi (elative plural)
kvasiryhmääsi (illative singular)
kvasiryhmiisi (illative plural)
kvasiryhmälläsi (adessive singular)
kvasiryhmilläsi (adessive plural)
kvasiryhmältäsi (ablative singular)
kvasiryhmiltäsi (ablative plural)
kvasiryhmällesi (allative singular)
kvasiryhmillesi (allative plural)
kvasiryhmänäsi (essive singular)
kvasiryhminäsi (essive plural)
kvasiryhmäksesi (singular translative)
kvasiryhmiksesi (plural translative)
kvasiryhmättäsi (abessive singular)
kvasiryhmittäsi (abessive plural)
kvasiryhminesi (comitative plural)
kvasiryhmämme (nominative singular)
kvasiryhmämme (nominative plural)
kvasiryhmämme (accusative nominative singular)
kvasiryhmämme (accusative nominative plural)
kvasiryhmämme (accusative genitive singular)
kvasiryhmämme (accusative genitive plural)
kvasiryhmämme (genitive singular)
kvasiryhmiemme (genitive plural)
kvasiryhmäimme (genitive plural rare)
kvasiryhmäämme (partitive singular)
kvasiryhmiämme (partitive plural)
kvasiryhmässämme (inessive singular)
kvasiryhmissämme (inessive plural)
kvasiryhmästämme (elative singular)
kvasiryhmistämme (elative plural)
kvasiryhmäämme (illative singular)
kvasiryhmiimme (illative plural)
kvasiryhmällämme (adessive singular)
kvasiryhmillämme (adessive plural)
kvasiryhmältämme (ablative singular)
kvasiryhmiltämme (ablative plural)
kvasiryhmällemme (allative singular)
kvasiryhmillemme (allative plural)
kvasiryhmänämme (essive singular)
kvasiryhminämme (essive plural)
kvasiryhmäksemme (singular translative)
kvasiryhmiksemme (plural translative)
kvasiryhmättämme (abessive singular)
kvasiryhmittämme (abessive plural)
kvasiryhminemme (comitative plural)
kvasiryhmänne (nominative singular)
kvasiryhmänne (nominative plural)
kvasiryhmänne (accusative nominative singular)
kvasiryhmänne (accusative nominative plural)
kvasiryhmänne (accusative genitive singular)
kvasiryhmänne (accusative genitive plural)
kvasiryhmänne (genitive singular)
kvasiryhmienne (genitive plural)
kvasiryhmäinne (genitive plural rare)
kvasiryhmäänne (partitive singular)
kvasiryhmiänne (partitive plural)
kvasiryhmässänne (inessive singular)
kvasiryhmissänne (inessive plural)
kvasiryhmästänne (elative singular)
kvasiryhmistänne (elative plural)
kvasiryhmäänne (illative singular)
kvasiryhmiinne (illative plural)
kvasiryhmällänne (adessive singular)
kvasiryhmillänne (adessive plural)
kvasiryhmältänne (ablative singular)
kvasiryhmiltänne (ablative plural)
kvasiryhmällenne (allative singular)
kvasiryhmillenne (allative plural)
kvasiryhmänänne (essive singular)
kvasiryhminänne (essive plural)
kvasiryhmäksenne (singular translative)
kvasiryhmiksenne (plural translative)
kvasiryhmättänne (abessive singular)
kvasiryhmittänne (abessive plural)
kvasiryhminenne (comitative plural)
kvasiryhmänsä (nominative singular)
kvasiryhmänsä (nominative plural)
kvasiryhmänsä (accusative nominative singular)
kvasiryhmänsä (accusative nominative plural)
kvasiryhmänsä (accusative genitive singular)
kvasiryhmänsä (accusative genitive plural)
kvasiryhmänsä (genitive singular)
kvasiryhmiensä (genitive plural)
kvasiryhmäinsä (genitive plural rare)
kvasiryhmäänsä (partitive singular)
kvasiryhmiään (partitive plural)
kvasiryhmiänsä (partitive plural)
kvasiryhmässään (inessive singular)
kvasiryhmässänsä (inessive singular)
kvasiryhmissään (inessive plural)
kvasiryhmissänsä (inessive plural)
kvasiryhmästään (elative singular)
kvasiryhmästänsä (elative singular)
kvasiryhmistään (elative plural)
kvasiryhmistänsä (elative plural)
kvasiryhmäänsä (illative singular)
kvasiryhmiinsä (illative plural)
kvasiryhmällään (adessive singular)
kvasiryhmällänsä (adessive singular)
kvasiryhmillään (adessive plural)
kvasiryhmillänsä (adessive plural)
kvasiryhmältään (ablative singular)
kvasiryhmältänsä (ablative singular)
kvasiryhmiltään (ablative plural)
kvasiryhmiltänsä (ablative plural)
kvasiryhmälleen (allative singular)
kvasiryhmällensä (allative singular)
kvasiryhmilleen (allative plural)
kvasiryhmillensä (allative plural)
kvasiryhmänään (essive singular)
kvasiryhmänänsä (essive singular)
kvasiryhminään (essive plural)
kvasiryhminänsä (essive plural)
kvasiryhmäkseen (singular translative)
kvasiryhmäksensä (singular translative)
kvasiryhmikseen (plural translative)
kvasiryhmiksensä (plural translative)
kvasiryhmättään (abessive singular)
kvasiryhmättänsä (abessive singular)
kvasiryhmittään (abessive plural)
kvasiryhmittänsä (abessive plural)
kvasiryhmineen (comitative plural)
kvasiryhminensä (comitative plural)

[ˈkʋɑ̝s̠iˌryçmæ]


neutraalialkio

neutraalialkio (nominative singular)
neutraalialkiot (nominative plural)
neutraalialkio (accusative nominative singular)
neutraalialkiot (accusative nominative plural)
neutraalialkion (accusative genitive singular)
neutraalialkiot (accusative genitive plural)
neutraalialkion (genitive singular)
neutraalialkioiden (genitive plural)
neutraalialkioitten (genitive plural)
neutraalialkiota (partitive singular)
neutraalialkioita (partitive plural)
neutraalialkiossa (inessive singular)
neutraalialkioissa (inessive plural)
neutraalialkiosta (elative singular)
neutraalialkioista (elative plural)
neutraalialkioon (illative singular)
neutraalialkioihin (illative plural)
neutraalialkiolla (adessive singular)
neutraalialkioilla (adessive plural)
neutraalialkiolta (ablative singular)
neutraalialkioilta (ablative plural)
neutraalialkiolle (allative singular)
neutraalialkioille (allative plural)
neutraalialkiona (essive singular)
neutraalialkioina (essive plural)
neutraalialkioksi (singular translative)
neutraalialkioiksi (plural translative)
neutraalialkiotta (abessive singular)
neutraalialkioitta (abessive plural)
neutraalialkioin (instructive plural)
See the possessive forms below. (comitative)
neutraalialkioni (nominative singular)
neutraalialkioni (nominative plural)
neutraalialkioni (accusative nominative singular)
neutraalialkioni (accusative nominative plural)
neutraalialkioni (accusative genitive singular)
neutraalialkioni (accusative genitive plural)
neutraalialkioni (genitive singular)
neutraalialkioideni (genitive plural)
neutraalialkioitteni (genitive plural)
neutraalialkiotani (partitive singular)
neutraalialkioitani (partitive plural)
neutraalialkiossani (inessive singular)
neutraalialkioissani (inessive plural)
neutraalialkiostani (elative singular)
neutraalialkioistani (elative plural)
neutraalialkiooni (illative singular)
neutraalialkioihini (illative plural)
neutraalialkiollani (adessive singular)
neutraalialkioillani (adessive plural)
neutraalialkioltani (ablative singular)
neutraalialkioiltani (ablative plural)
neutraalialkiolleni (allative singular)
neutraalialkioilleni (allative plural)
neutraalialkionani (essive singular)
neutraalialkioinani (essive plural)
neutraalialkiokseni (singular translative)
neutraalialkioikseni (plural translative)
neutraalialkiottani (abessive singular)
neutraalialkioittani (abessive plural)
neutraalialkioineni (comitative plural)
neutraalialkiosi (nominative singular)
neutraalialkiosi (nominative plural)
neutraalialkiosi (accusative nominative singular)
neutraalialkiosi (accusative nominative plural)
neutraalialkiosi (accusative genitive singular)
neutraalialkiosi (accusative genitive plural)
neutraalialkiosi (genitive singular)
neutraalialkioidesi (genitive plural)
neutraalialkioittesi (genitive plural)
neutraalialkiotasi (partitive singular)
neutraalialkioitasi (partitive plural)
neutraalialkiossasi (inessive singular)
neutraalialkioissasi (inessive plural)
neutraalialkiostasi (elative singular)
neutraalialkioistasi (elative plural)
neutraalialkioosi (illative singular)
neutraalialkioihisi (illative plural)
neutraalialkiollasi (adessive singular)
neutraalialkioillasi (adessive plural)
neutraalialkioltasi (ablative singular)
neutraalialkioiltasi (ablative plural)
neutraalialkiollesi (allative singular)
neutraalialkioillesi (allative plural)
neutraalialkionasi (essive singular)
neutraalialkioinasi (essive plural)
neutraalialkioksesi (singular translative)
neutraalialkioiksesi (plural translative)
neutraalialkiottasi (abessive singular)
neutraalialkioittasi (abessive plural)
neutraalialkioinesi (comitative plural)
neutraalialkiomme (nominative singular)
neutraalialkiomme (nominative plural)
neutraalialkiomme (accusative nominative singular)
neutraalialkiomme (accusative nominative plural)
neutraalialkiomme (accusative genitive singular)
neutraalialkiomme (accusative genitive plural)
neutraalialkiomme (genitive singular)
neutraalialkioidemme (genitive plural)
neutraalialkioittemme (genitive plural)
neutraalialkiotamme (partitive singular)
neutraalialkioitamme (partitive plural)
neutraalialkiossamme (inessive singular)
neutraalialkioissamme (inessive plural)
neutraalialkiostamme (elative singular)
neutraalialkioistamme (elative plural)
neutraalialkioomme (illative singular)
neutraalialkioihimme (illative plural)
neutraalialkiollamme (adessive singular)
neutraalialkioillamme (adessive plural)
neutraalialkioltamme (ablative singular)
neutraalialkioiltamme (ablative plural)
neutraalialkiollemme (allative singular)
neutraalialkioillemme (allative plural)
neutraalialkionamme (essive singular)
neutraalialkioinamme (essive plural)
neutraalialkioksemme (singular translative)
neutraalialkioiksemme (plural translative)
neutraalialkiottamme (abessive singular)
neutraalialkioittamme (abessive plural)
neutraalialkioinemme (comitative plural)
neutraalialkionne (nominative singular)
neutraalialkionne (nominative plural)
neutraalialkionne (accusative nominative singular)
neutraalialkionne (accusative nominative plural)
neutraalialkionne (accusative genitive singular)
neutraalialkionne (accusative genitive plural)
neutraalialkionne (genitive singular)
neutraalialkioidenne (genitive plural)
neutraalialkioittenne (genitive plural)
neutraalialkiotanne (partitive singular)
neutraalialkioitanne (partitive plural)
neutraalialkiossanne (inessive singular)
neutraalialkioissanne (inessive plural)
neutraalialkiostanne (elative singular)
neutraalialkioistanne (elative plural)
neutraalialkioonne (illative singular)
neutraalialkioihinne (illative plural)
neutraalialkiollanne (adessive singular)
neutraalialkioillanne (adessive plural)
neutraalialkioltanne (ablative singular)
neutraalialkioiltanne (ablative plural)
neutraalialkiollenne (allative singular)
neutraalialkioillenne (allative plural)
neutraalialkionanne (essive singular)
neutraalialkioinanne (essive plural)
neutraalialkioksenne (singular translative)
neutraalialkioiksenne (plural translative)
neutraalialkiottanne (abessive singular)
neutraalialkioittanne (abessive plural)
neutraalialkioinenne (comitative plural)
neutraalialkionsa (nominative singular)
neutraalialkionsa (nominative plural)
neutraalialkionsa (accusative nominative singular)
neutraalialkionsa (accusative nominative plural)
neutraalialkionsa (accusative genitive singular)
neutraalialkionsa (accusative genitive plural)
neutraalialkionsa (genitive singular)
neutraalialkioidensa (genitive plural)
neutraalialkioittensa (genitive plural)
neutraalialkiotaan (partitive singular)
neutraalialkiotansa (partitive singular)
neutraalialkioitaan (partitive plural)
neutraalialkioitansa (partitive plural)
neutraalialkiossaan (inessive singular)
neutraalialkiossansa (inessive singular)
neutraalialkioissaan (inessive plural)
neutraalialkioissansa (inessive plural)
neutraalialkiostaan (elative singular)
neutraalialkiostansa (elative singular)
neutraalialkioistaan (elative plural)
neutraalialkioistansa (elative plural)
neutraalialkioonsa (illative singular)
neutraalialkioihinsa (illative plural)
neutraalialkiollaan (adessive singular)
neutraalialkiollansa (adessive singular)
neutraalialkioillaan (adessive plural)
neutraalialkioillansa (adessive plural)
neutraalialkioltaan (ablative singular)
neutraalialkioltansa (ablative singular)
neutraalialkioiltaan (ablative plural)
neutraalialkioiltansa (ablative plural)
neutraalialkiolleen (allative singular)
neutraalialkiollensa (allative singular)
neutraalialkioilleen (allative plural)
neutraalialkioillensa (allative plural)
neutraalialkionaan (essive singular)
neutraalialkionansa (essive singular)
neutraalialkioinaan (essive plural)
neutraalialkioinansa (essive plural)
neutraalialkiokseen (singular translative)
neutraalialkioksensa (singular translative)
neutraalialkioikseen (plural translative)
neutraalialkioiksensa (plural translative)
neutraalialkiottaan (abessive singular)
neutraalialkiottansa (abessive singular)
neutraalialkioittaan (abessive plural)
neutraalialkioittansa (abessive plural)
neutraalialkioineen (comitative plural)
neutraalialkioinensa (comitative plural)

(algebra) identity (such as an additive or multiplicative identity)


Hurwitz algebra

Hurwitz algebra (plural Hurwitz algebras)

Named after German mathematician Adolf Hurwitz (1859–1919); see also Hurwitz problem, Hurwitz's theorem.

As we recall, a Hurwitz algebra is a composition algebra with unit element.

1995, Susumo Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, page 40:

Observe that the unital composition algebra associated with (C,#92;star) by the construction given in the proof of Proposition (33.27) is the Hurwitz algebra (C,#92;diamond) if we set a#61;1.

1998, Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol, The Book of Involutions, American Mathematical Society, page 465:

2006, Alberto Elduque, Chapter 12: A new look at Freudenthal's Magic Square, Lev Sabinin, Larissa Sbitneva, Ivan Shestakov (editors, Non-Associative Algebra and Its Applications, Taylor & Francis Group (Chapman & Hall/CRC), page 15,
Unital composition algebras (or Hurwitz algebras) form a well-known class of algebras. Any Hurwitz algebra has a finite dimension equal to either 1, 2, 4, or 8. The two-dimensional Hurwitz algebras are the quadratic étale algebras over the ground field F, the four dimensional ones are the generalized quaternion algebras, and the eight dimensional Hurwitz algebras are called Cayley algebras, and are analog to the classical algebra of octonions (for a good survey of the latter, see [Bae02]).

(algebra) Any one of the unital composition algebras identified by Hurwitz's theorem (on composition algebras) as solutions to the Hurwitz problem.


hệ số

[hej˨˩ʔ ʂow˨˩˦]

Sino-Vietnamese word from 係數, from Japanese 係数 (keisū).


yhtälöpari

yhtälöpari (nominative singular)
yhtälöparit (nominative plural)
yhtälöpari (accusative nominative singular)
yhtälöparit (accusative nominative plural)
yhtälöparin (accusative genitive singular)
yhtälöparit (accusative genitive plural)
yhtälöparin (genitive singular)
yhtälöparien (genitive plural)
yhtälöparia (partitive singular)
yhtälöpareja (partitive plural)
yhtälöparissa (inessive singular)
yhtälöpareissa (inessive plural)
yhtälöparista (elative singular)
yhtälöpareista (elative plural)
yhtälöpariin (illative singular)
yhtälöpareihin (illative plural)
yhtälöparilla (adessive singular)
yhtälöpareilla (adessive plural)
yhtälöparilta (ablative singular)
yhtälöpareilta (ablative plural)
yhtälöparille (allative singular)
yhtälöpareille (allative plural)
yhtälöparina (essive singular)
yhtälöpareina (essive plural)
yhtälöpariksi (singular translative)
yhtälöpareiksi (plural translative)
yhtälöparitta (abessive singular)
yhtälöpareitta (abessive plural)
yhtälöparein (instructive plural)
See the possessive forms below. (comitative)
yhtälöparini (nominative singular)
yhtälöparini (nominative plural)
yhtälöparini (accusative nominative singular)
yhtälöparini (accusative nominative plural)
yhtälöparini (accusative genitive singular)
yhtälöparini (accusative genitive plural)
yhtälöparini (genitive singular)
yhtälöparieni (genitive plural)
yhtälöpariani (partitive singular)
yhtälöparejani (partitive plural)
yhtälöparissani (inessive singular)
yhtälöpareissani (inessive plural)
yhtälöparistani (elative singular)
yhtälöpareistani (elative plural)
yhtälöpariini (illative singular)
yhtälöpareihini (illative plural)
yhtälöparillani (adessive singular)
yhtälöpareillani (adessive plural)
yhtälöpariltani (ablative singular)
yhtälöpareiltani (ablative plural)
yhtälöparilleni (allative singular)
yhtälöpareilleni (allative plural)
yhtälöparinani (essive singular)
yhtälöpareinani (essive plural)
yhtälöparikseni (singular translative)
yhtälöpareikseni (plural translative)
yhtälöparittani (abessive singular)
yhtälöpareittani (abessive plural)
yhtälöpareineni (comitative plural)
yhtälöparisi (nominative singular)
yhtälöparisi (nominative plural)
yhtälöparisi (accusative nominative singular)
yhtälöparisi (accusative nominative plural)
yhtälöparisi (accusative genitive singular)
yhtälöparisi (accusative genitive plural)
yhtälöparisi (genitive singular)
yhtälöpariesi (genitive plural)
yhtälöpariasi (partitive singular)
yhtälöparejasi (partitive plural)
yhtälöparissasi (inessive singular)
yhtälöpareissasi (inessive plural)
yhtälöparistasi (elative singular)
yhtälöpareistasi (elative plural)
yhtälöpariisi (illative singular)
yhtälöpareihisi (illative plural)
yhtälöparillasi (adessive singular)
yhtälöpareillasi (adessive plural)
yhtälöpariltasi (ablative singular)
yhtälöpareiltasi (ablative plural)
yhtälöparillesi (allative singular)
yhtälöpareillesi (allative plural)
yhtälöparinasi (essive singular)
yhtälöpareinasi (essive plural)
yhtälöpariksesi (singular translative)
yhtälöpareiksesi (plural translative)
yhtälöparittasi (abessive singular)
yhtälöpareittasi (abessive plural)
yhtälöpareinesi (comitative plural)
yhtälöparimme (nominative singular)
yhtälöparimme (nominative plural)
yhtälöparimme (accusative nominative singular)
yhtälöparimme (accusative nominative plural)
yhtälöparimme (accusative genitive singular)
yhtälöparimme (accusative genitive plural)
yhtälöparimme (genitive singular)
yhtälöpariemme (genitive plural)
yhtälöpariamme (partitive singular)
yhtälöparejamme (partitive plural)
yhtälöparissamme (inessive singular)
yhtälöpareissamme (inessive plural)
yhtälöparistamme (elative singular)
yhtälöpareistamme (elative plural)
yhtälöpariimme (illative singular)
yhtälöpareihimme (illative plural)
yhtälöparillamme (adessive singular)
yhtälöpareillamme (adessive plural)
yhtälöpariltamme (ablative singular)
yhtälöpareiltamme (ablative plural)
yhtälöparillemme (allative singular)
yhtälöpareillemme (allative plural)
yhtälöparinamme (essive singular)
yhtälöpareinamme (essive plural)
yhtälöpariksemme (singular translative)
yhtälöpareiksemme (plural translative)
yhtälöparittamme (abessive singular)
yhtälöpareittamme (abessive plural)
yhtälöpareinemme (comitative plural)
yhtälöparinne (nominative singular)
yhtälöparinne (nominative plural)
yhtälöparinne (accusative nominative singular)
yhtälöparinne (accusative nominative plural)
yhtälöparinne (accusative genitive singular)
yhtälöparinne (accusative genitive plural)
yhtälöparinne (genitive singular)
yhtälöparienne (genitive plural)
yhtälöparianne (partitive singular)
yhtälöparejanne (partitive plural)
yhtälöparissanne (inessive singular)
yhtälöpareissanne (inessive plural)
yhtälöparistanne (elative singular)
yhtälöpareistanne (elative plural)
yhtälöpariinne (illative singular)
yhtälöpareihinne (illative plural)
yhtälöparillanne (adessive singular)
yhtälöpareillanne (adessive plural)
yhtälöpariltanne (ablative singular)
yhtälöpareiltanne (ablative plural)
yhtälöparillenne (allative singular)
yhtälöpareillenne (allative plural)
yhtälöparinanne (essive singular)
yhtälöpareinanne (essive plural)
yhtälöpariksenne (singular translative)
yhtälöpareiksenne (plural translative)
yhtälöparittanne (abessive singular)
yhtälöpareittanne (abessive plural)
yhtälöpareinenne (comitative plural)
yhtälöparinsa (nominative singular)
yhtälöparinsa (nominative plural)
yhtälöparinsa (accusative nominative singular)
yhtälöparinsa (accusative nominative plural)
yhtälöparinsa (accusative genitive singular)
yhtälöparinsa (accusative genitive plural)
yhtälöparinsa (genitive singular)
yhtälöpariensa (genitive plural)
yhtälöpariaan (partitive singular)
yhtälöpariansa (partitive singular)
yhtälöparejaan (partitive plural)
yhtälöparejansa (partitive plural)
yhtälöparissaan (inessive singular)
yhtälöparissansa (inessive singular)
yhtälöpareissaan (inessive plural)
yhtälöpareissansa (inessive plural)
yhtälöparistaan (elative singular)
yhtälöparistansa (elative singular)
yhtälöpareistaan (elative plural)
yhtälöpareistansa (elative plural)
yhtälöpariinsa (illative singular)
yhtälöpareihinsa (illative plural)
yhtälöparillaan (adessive singular)
yhtälöparillansa (adessive singular)
yhtälöpareillaan (adessive plural)
yhtälöpareillansa (adessive plural)
yhtälöpariltaan (ablative singular)
yhtälöpariltansa (ablative singular)
yhtälöpareiltaan (ablative plural)
yhtälöpareiltansa (ablative plural)
yhtälöparilleen (allative singular)
yhtälöparillensa (allative singular)
yhtälöpareilleen (allative plural)
yhtälöpareillensa (allative plural)
yhtälöparinaan (essive singular)
yhtälöparinansa (essive singular)
yhtälöpareinaan (essive plural)
yhtälöpareinansa (essive plural)
yhtälöparikseen (singular translative)
yhtälöpariksensa (singular translative)
yhtälöpareikseen (plural translative)
yhtälöpareiksensa (plural translative)
yhtälöparittaan (abessive singular)
yhtälöparittansa (abessive singular)
yhtälöpareittaan (abessive plural)
yhtälöpareittansa (abessive plural)
yhtälöpareineen (comitative plural)
yhtälöpareinensa (comitative plural)

[ˈyçt̪ælø̞ˌpɑ̝ri]

system of two equations
system of equations

(algebra) pair of equations, two simultaneous equations, system of two equations


integral element

integral element (plural integral elements)

1956, Unnamed translator, D. K Faddeev, Simple Algebras Over a Field of Algebraic Functions of One Variable, in Five Papers on Logic Algebra, and Number Theory, American Mathematical Society Translations, Series 2, Volume 3, page 21,
A subring of 𝔅 containing the ring o of integral elements of the field k_0(π), distinct from 𝔅, and not contained in any other subring of 𝔅 distinct from 𝔅, is called a maximal ring of the algebra 𝔅. In a division algebra, the only maximal ring is the ring of integral elements.

1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, 2003 Softcover Reprint, page 172,
If 𝔖 is the ring of integral elements in a commutative ring 𝔗 (over a subring R) and if the element t of 𝔗 is integral over 𝔖, then t is also integral over R (that is, contained in 𝔖).

In this paper the notion of the ring of all integral elements of a number field was put in the central place of his^([Richard Dedekind's]) theory.

2004, Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko, Algebras, Rings and Modules, Volume 1, Kluwer Academic Publishers, page 209:

(algebra, commutative algebra, ring theory) Given a commutative unital ring R with extension ring S (i.e., that is a subring of S), any element s ∈ S that is a root of some monic polynomial with coefficients in R.

elemento intero

element of a given ring that is a root of some monic polynomial with coefficients in a given subring


สมการ

สม (sǒm) + การ (gaan). Compare similar Sanskrit समीकार (samīkāra).


ညီမျှခြင်း

ညီ မျှ ခြင်း (canonical)
nyihmya.hkrang: (romanization)

/ɲìm̥ja̰d͡ʑɪ́ɴ/


algebraic poset

algebraic poset (plural algebraic posets)

1985 October, Rudolf-E. Hoffmann, The Injective Hull and the 𝒞ℒ-Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833,
A poset (P,<) is said to be algebraic if and only if
i) P is up-complete, i.e., for every non-empty up-directed subset D,thesupremum operatorname supd exists,
ii) for every x∈P, the set
K_X:=y∈P|ycompact,y<x
is non-empty and up-directed, and
x= operatorname supK_x.
A poset P is an algebraic poset if and only if it is a continuous poset in which, for every x,y∈P,x≪y (if and) only if x<c<y for some compact element c of P.
Concerning the definition of an algebraic poset, a caveat may be in order (which, mutatis mutandis, applies to continuous posets): it may happen that all of the axioms for an algebraic poset are satisfied except that the sets K_x fail to be up-directed ([50], 4.2 or [49], 4.5). Even when "enough" compact sets are readily available, it sometimes remains a delicate problem to verify the up-directedness of the sets K_x.
The concept of an algebraic poset arose in theoretical computer science ([50], [54], cf. also [14]). It is a natural extension of he familiar notion of a (complete) "algebraic lattice" (cf. [9], [20], I-4).

ALG is the category whose objects are algebraic posets and whose morphisms are continuous functions.
A structure (D,#92;le#95;D,#92;cup#95;D) where (D,#92;le#95;D) is an algebraic poset and #92;cup#95;D is a binary operation on D which is continuous (in both variables), commutative, associative and absorptive (i.e., d#92;cup#95;Dd#61;d for all d#92;inD) will be called a nondeterministic algebraic poset.

1988, Karel Hrbacek, “A Powerdomain Construction”, in Michael Main, Austin Melton, Michael Mislove, David Schmidt, editors, Mathematical Foundations of Programming Language Semantics: 3rd Workshop, Proceedings, Springer, page 202:

It is important to note that we have not assumed an algebraic poset is a cpo. The canonical example of an algebraic poset is #123;A#42;#92;cupA#42;#125;#92;sqrt#123;#125; in the prefix order; this poset enjoys the added condition that every element is compact.

1992, Stephen D. Brookes et al., editors, Mathematical Foundations of Programming Semantics: 7th International Conference, Proceedings, Springer, page 88:

The correspondence between primes and occurrences suggests that given an abstract prime algebraic poset, we may construct a behavioural presentation from it.

1997, Michael W. Shields, Semantics of Parallelism: Non-Interleaving Representation of Behaviour, Springer, page 41:

(algebra, order theory) A partially ordered set (poset) in which every element is the supremum of the compact elements below it.