Simple ring

A ring, containing more than one element, without two-sided ideals (cf. Ideal) different from 0 and the entire ring. An associative simple ring with an identity element and containing a minimal one-sided ideal is isomorphic to a matrix ring over a some skew-field (cf. also Associative rings and algebras). Without the assumption on the existence of an identity, such a ring is locally matrix over some skew-field $D$, i.e. each of its finite subsets is contained in a subring isomorphic to a matrix ring over $D$ (cf. [2]). There are simple rings without zero divisors (even Noetherian simple rings, cf. also Noetherian ring) different from a skew-field, as well as Noetherian simple rings with zero divisors but without idempotents [3]. Simple rings that are radical in the sense of N. Jacobson are known (cf. [1]). Simple nil rings were constructed by Smoktunowicz in 2002 (cf. [6]).