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A ring, containing more than one element, without two-sided ideals (cf. [[Ideal|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|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. [[#References|[2]]]). There are simple rings without zero divisors (even Noetherian simple rings, cf. also [[Noetherian ring|Noetherian ring]]) different from a skew-field, as well as Noetherian simple rings with zero divisors but without idempotents [[#References|[3]]]. Simple rings that are radical in the sense of N. Jacobson are known (cf. [[#References|[1]]]). Simple nil rings were constructed by Smoktunowicz in 2002 (cf. [[#References|[6]]]).



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

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Bokut',   "Associative rings" , '''1–2''' , Novosibirsk  (1977–1981)  (In Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,   "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  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)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  C. Faith,   "Algebra" , '''1–2''' , Springer  (1973–1976)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  J. Cozzens,   C. Faith,   "Simple Noetherian rings" , Cambridge Univ. Press  (1975)</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">A. Smoktunowicz,   "A Simple Nil Ring Exists", Comm. in Algebra  30 (2002), pp. 27-59.</TD></TR>
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