prime ring

English

Noun

prime ring (plural prime rings)

2. (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^[1], 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.

3. (algebra, ring theory) Synonym of prime subring
   • 2012, Sebastian Xambo-Descamps, Block Error-Correcting Codes: A Computational Primer, Springer Science & Business Media, page 110:

     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.

   2. (algebra, ring theory, uncommon) A ring which is equal to its own prime subring.
      • 2012, Thomas Becker, Volker Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra, Springer Science & Business Media, page 50:

        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.

      • 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:

        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 ℤ_n.

Usage notes

The following conditions are equivalent to R being a prime ring (sense 1):

• 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.

Sense 1 and sense 2.1 are not equivalent; for example, $\mathbb {Z} [x]$ is a prime ring in the sense of sense 1, but not in the sense of sense 2.1.

See also

• annihilator
• prime ideal
• semiprime ring

Further reading

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

Anagrams

• repriming