View source for Semi-simple algebra

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''with respect to a radical $r$''

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.



====Comments====
By Wedderburn's theorem (cf. [[Wedderburn–Artin theorem|Wedderburn–Artin theorem]]), an Artinian algebra with [[Jacobson radical|Jacobson radical]] zero is a finite direct sum of simple algebras.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Algebra" , '''2''' , Wiley  (1989)  pp. Chapt. 5</TD></TR></table>