prime ring

prime ring

English

Noun

prime ring (plural prime rings)

1. (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.
   • 1969, Taita Journal of Mathematics, Volumes 1-2, page 56,

     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.

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

     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.

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

     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.

Usage notes

Equivalently, either of the following conditions determine that ring R is a prime ring:

• for arbitrary a, b ∈ R, if arb = 0 for all r ∈ R (i.e., if aRb = 0) then either a = 0 or b = 0;
• the zero ideal is a prime ideal in R.

See also

• annihilator
• prime ideal
• semiprime ring

Further reading

• Semiprime ring on Wikipedia.Wikipedia
• Prime ring on Encyclopedia of Mathematics

Anagrams

• repriming