Semisimple algebra

In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). 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. The radical contains all nilpotent ideals, and if the algebra is finite-dimensional, the radical itself is a nilpotent ideal. A finite-dimensional algebra is then said to be semisimple if its radical contains only the zero element.

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. The only possible ideals of a simple algebra A are A and {0}. Thus if A is not nilpotent, then A is semisimple. Because A² is an ideal of A and A is simple, A² = A. By induction, Aⁿ = A for every positive integer n, i.e. A is not nilpotent.

Any self-adjoint subalgebra A of n × n matrices with complex entries is semisimple. Let Rad(A) be the radical of A. Suppose a matrix M is in Rad(A). 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. So M x is the zero vector for all x, i.e. M = 0.

If {A_i} is a finite collection of simple algebras, then their Cartesian product ∏ A_i is semisimple. If (a_i) 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 ∏ A_i. This implies, for all b in A₁, a₁b is nilpotent in A₁, i.e. a₁ ∈ Rad(A₁). So a₁ = 0. Similarly, a_i = 0 for all other i.

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