Difference between revisions of "Semi-simple algebra"
From Encyclopedia of Mathematics
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| − | ''with respect to a radical | + | {{TEX|done}} |
| + | ''with respect to a radical $r$'' | ||
| − | An algebra which is an | + | An algebra which is an $r$-semi-simple ring (see [[Semi-simple ring|Semi-simple ring]]). In some classes of algebras and for a suitable choice of the radical $r$, it is possible to describe the structure of a semi-simple algebra (see [[Classical semi-simple ring|Classical semi-simple ring]]; [[Alternative rings and algebras|Alternative rings and algebras]]; [[Jordan algebra|Jordan algebra]]; [[Lie algebra, semi-simple|Lie algebra, semi-simple]]). |
By a semi-simple algebra one frequently understands a finite-dimensional algebra over a field which is a direct sum of simple algebras. | By a semi-simple algebra one frequently understands a finite-dimensional algebra over a field which is a direct sum of simple algebras. | ||
Latest revision as of 16:31, 5 August 2014
with respect to a radical $r$
An algebra which is an $r$-semi-simple ring (see Semi-simple ring). In some classes of algebras and for a suitable choice of the radical $r$, it is possible to describe the structure of a semi-simple algebra (see Classical semi-simple ring; Alternative rings and algebras; Jordan algebra; Lie algebra, semi-simple).
By a semi-simple algebra one frequently understands a finite-dimensional algebra over a field which is a direct sum of simple algebras.
Comments
By Wedderburn's theorem (cf. Wedderburn–Artin theorem), an Artinian algebra with Jacobson radical zero is a finite direct sum of simple algebras.
References
| [a1] | P.M. Cohn, "Algebra" , 2 , Wiley (1989) pp. Chapt. 5 |
How to Cite This Entry:
Semi-simple algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-simple_algebra&oldid=17192
Semi-simple algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-simple_algebra&oldid=17192
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article