prime ring meaning: definition, quotations

prime ring

noun

Algebra, Mathematics, Sciences 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:

1 more example

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, Mathematics, Sciences 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:

Algebra, Mathematics, Sciences Uncommon 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:

Related terms: annihilator, prime ideal, semiprime ring