See prime ring in All languages combined, or Wiktionary
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It is obvious that a prime ring is necessarily semi-prime.", "type": "quote" }, { "ref": "1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products, Elsevier (North-Holland), page 223:", "text": "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.", "type": "quote" }, { "ref": "2014, Matej Brešar, Introduction to Noncommutative Algebra, Springer, page 163:", "text": "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.", "type": "quote" } ], "glosses": [ "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." ], "id": "en-prime_ring-en-noun-en:productofideals", "links": [ [ "algebra", "algebra" ], [ "ring theory", "ring theory" ], [ "ring", "ring" ], [ "ideal", "ideal" ] ], "qualifier": "ring theory", "raw_glosses": [ "(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." ], "senseid": [ "en:productofideals" ], "topics": [ "algebra", "mathematics", "sciences" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Algebra", "orig": "en:Algebra", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "16 42 42", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "19 41 41", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "19 40 40", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" } ], "examples": [ { "ref": "2012, Sebastian Xambo-Descamps, Block Error-Correcting Codes: A Computational Primer, Springer Science & Business Media, page 110:", "text": "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.", "type": "quote" } ], "glosses": [ "Synonym of prime subring" ], "id": "en-prime_ring-en-noun-g-xU33sk", "links": [ [ "algebra", "algebra" ], [ "ring theory", "ring theory" ], [ "prime subring", "prime subring#English" ] ], "qualifier": "ring theory", "raw_glosses": [ "(algebra, ring theory) Synonym of prime subring" ], "synonyms": [ { "tags": [ "synonym", "synonym-of" ], "word": "prime subring" } ], "topics": [ "algebra", "mathematics", "sciences" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Algebra", "orig": "en:Algebra", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "16 42 42", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "19 41 41", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "19 40 40", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" } ], "examples": [ { "ref": "2012, Thomas Becker, Volker Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra, Springer Science & Business Media, page 50:", "text": "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.", "type": "quote" }, { "ref": "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:", "text": "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.", "type": "quote" } ], "glosses": [ "Synonym of prime subring", "A ring which is equal to its own prime subring." ], "id": "en-prime_ring-en-noun-en:ownprimesubring", "links": [ [ "algebra", "algebra" ], [ "ring theory", "ring theory" ], [ "prime subring", "prime subring#English" ], [ "ring", "ring" ], [ "prime subring", "prime subring" ] ], "qualifier": "ring theory", "raw_glosses": [ "(algebra, ring theory) Synonym of prime subring", "(algebra, ring theory, uncommon) A ring which is equal to its own prime subring." ], "related": [ { "_dis1": "21 26 53", "word": "annihilator" }, { "_dis1": "21 26 53", "word": "prime ideal" }, { "_dis1": "21 26 53", "word": "semiprime ring" } ], "senseid": [ "en:ownprimesubring" ], "tags": [ "uncommon" ], "topics": [ "algebra", "mathematics", "sciences" ] } ], "wikipedia": [ "prime ring" ], "word": "prime ring" }
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This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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