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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085290/s0852901.png" />, i.e. each of its finite subsets is contained in a subring isomorphic to a matrix ring over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085290/s0852902.png" /> (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]]]). However, the problem of the existence of simple nil rings remains open.
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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]]]).
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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]].
 
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====
 
====References====
<table><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></table>
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<table>
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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>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR>
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<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>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  C. Faith,  "Algebra" , '''1–2''' , Springer  (1973–1976)</TD></TR>
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<TR><TD valign="top">[5]</TD> <TD valign="top">  J. Cozzens,  C. Faith,  "Simple Noetherian rings" , Cambridge Univ. Press  (1975)</TD></TR>
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<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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</table>

Latest revision as of 09:55, 26 November 2016

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.
How to Cite This Entry:
Simple ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simple_ring&oldid=18150
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article