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]).

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

References

[1]	L.A. Bokut', "Associative rings" , 1–2 , Novosibirsk (1977–1981) (In Russian)
[2]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[3]	A.E. Zalesskii, O. Neroslavskii, "There exist simple Noetherian rings with zero division but without idempotents" Comm. in Algebra , 5 : 3 (1977) pp. 231–244 (In Russian) (English abstract)
[4]	C. Faith, "Algebra" , 1–2 , Springer (1973–1976)
[5]	J. Cozzens, C. Faith, "Simple Noetherian rings" , Cambridge Univ. Press (1975)
[6]	A. Smoktunowicz, "A Simple Nil Ring Exists", Comm. in Algebra 30 (2002), pp. 27-59.