Difference between revisions of "Wedderburn-Artin theorem"
From Encyclopedia of Mathematics
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| − | A theorem which fully describes the structure of associative | + | A theorem which fully describes the structure of associative [[Artinian ring]]s without nilpotent ideals. An [[Associative rings and algebras|associative ring]] $R$ has the minimum condition for right ideals and has no nilpotent ideals if and only if $R$ is the direct sum of a finite number of ideals, each of which is isomorphic to a complete [[matrix ring]] of finite order over a suitable skew-field; this decomposition into a direct sum is unique apart from the ordering of its terms. This theorem was first obtained by J. Wedderburn for finite-dimensional algebras over a field, and was proved by E. Artin [[#References|[1]]] in its final formulation. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Artin, "The influence of J.H.M. Wedderburn on the development of modern algebra" ''Bull. Amer. Math. Soc.'' , '''56''' (1950) pp. 65–72</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> E. Artin, "The influence of J.H.M. Wedderburn on the development of modern algebra" ''Bull. Amer. Math. Soc.'' , '''56''' (1950) pp. 65–72</TD></TR> | ||
| + | </table> | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.H.M. Wedderburn, "Lectures on matrices" , Dover, reprint (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C. Faith, "Algebra: rings, modules, and categories" , '''1''' , Springer (1973) pp. 380, 369</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P.M. Cohn, "Algebra" , '''2''' , Wiley (1989) pp. 174ff</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> J.H.M. Wedderburn, "Lectures on matrices" , Dover, reprint (1964)</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> C. Faith, "Algebra: rings, modules, and categories" , '''1''' , Springer (1973) pp. 380, 369</TD></TR> | ||
| + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> P.M. Cohn, "Algebra" , '''2''' , Wiley (1989) pp. 174ff</TD></TR> | ||
| + | </table> | ||
Latest revision as of 06:17, 13 September 2016
A theorem which fully describes the structure of associative Artinian rings without nilpotent ideals. An associative ring $R$ has the minimum condition for right ideals and has no nilpotent ideals if and only if $R$ is the direct sum of a finite number of ideals, each of which is isomorphic to a complete matrix ring of finite order over a suitable skew-field; this decomposition into a direct sum is unique apart from the ordering of its terms. This theorem was first obtained by J. Wedderburn for finite-dimensional algebras over a field, and was proved by E. Artin [1] in its final formulation.
References
| [1] | E. Artin, "The influence of J.H.M. Wedderburn on the development of modern algebra" Bull. Amer. Math. Soc. , 56 (1950) pp. 65–72 |
Comments
References
| [a1] | J.H.M. Wedderburn, "Lectures on matrices" , Dover, reprint (1964) |
| [a2] | C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) pp. 380, 369 |
| [a3] | P.M. Cohn, "Algebra" , 2 , Wiley (1989) pp. 174ff |
How to Cite This Entry:
Wedderburn-Artin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wedderburn-Artin_theorem&oldid=31724
Wedderburn-Artin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wedderburn-Artin_theorem&oldid=31724
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article