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.
|[a1]||P.M. Cohn, "Algebra" , 2 , Wiley (1989) pp. Chapt. 5|
Semi-simple algebra. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Semi-simple_algebra&oldid=32731