If the algebra is finite-dimensional this is equivalent to saying that it can be expressed as a Cartesian product of simple subalgebras.
The Jacobson radical of an algebra over a field is the ideal consisting of all elements that annihilate every simple left-module.
An algebra A is called simple if it has no proper ideals and A2 = {ab | a, b ∈ A} ≠ {0}.
As the terminology suggests, simple algebras are semisimple.
Any self-adjoint subalgebra A of n × n matrices with complex entries is semisimple.
Then M*M lies in some nilpotent ideals of A, therefore (M*M)k = 0 for some positive integer k. By positive-semidefiniteness of M*M, this implies M*M = 0.
If (ai) is an element of Rad(A) and e1 is the multiplicative identity in A1 (all simple algebras possess a multiplicative identity), then (a1, a2, ...) · (e1, 0, ...) = (a1, 0..., 0) lies in some nilpotent ideal of Π Ai.
It is less apparent from the definition that the converse of the above is also true, that is, any finite-dimensional semisimple algebra is isomorphic to a Cartesian product of a finite number of simple algebras.
First, invoking the assumption that A is semisimple, one can show that the J1 is a simple algebra (therefore unital).
Therefore, one can decompose By maximality of J1 as an ideal in J2 and also the semisimplicity of A, the algebra is simple.
Proceed by induction in similar fashion proves the claim.
For example, J3 is the Cartesian product of simple algebras The above result can be restated in a different way.
For a semisimple algebra A = A1 ×...× An expressed in terms of its simple factors, consider the units ei ∈ Ai.
Therefore, for every semisimple algebra A, there exists idempotents {Ei} in the center of A, such that A theorem due to Joseph Wedderburn completely classifies finite-dimensional semisimple algebras over a field
[1] This theorem was later generalized by Emil Artin to semisimple rings.