Semi-simple algebra

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