Namespaces
Variants
Actions

Difference between revisions of "Semi-simple algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
''with respect to a radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084330/s0843301.png" />''
+
{{TEX|done}}
 +
''with respect to a radical $r$''
  
An algebra which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084330/s0843302.png" />-semi-simple ring (see [[Semi-simple ring|Semi-simple ring]]). In some classes of algebras and for a suitable choice of the radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084330/s0843303.png" />, 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]]).
+
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
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article