# Simple ring

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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. ). 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 . Simple rings that are radical in the sense of N. Jacobson are known (cf. ). Simple nil rings were constructed by Smoktunowicz in 2002 (cf. ).

The description of the structure of alternative simple rings reduces to the associative case (cf. Alternative rings and algebras). See also Simple algebra.

How to Cite This Entry:
Simple ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simple_ring&oldid=39825
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article