Dictionary

prime ring en · NOUN

Meanings

1. (qualifier: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.
   • 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:
2. (qualifier:ring theory) Synonym of prime subring.
   • 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:
3. (qualifier:ring theory, uncommon) Synonym of prime subring. ▸ A ring which is equal to its own 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:

Forms

Relateds

semiprime ring · annihilator · prime ideal

Synonyms

prime subring