prime ring

== English == === Noun === prime ring (plural prime rings) (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. (algebra, ring theory) Synonym of prime subring. (algebra, ring theory, uncommon) A ring which is equal to its own prime subring. ==== 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, Z [ x ] {\displaystyle \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