# 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.

## Definition

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 A$^2$ = {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 simple, then A is not nilpotent. Because A$^2$ is an ideal of A and A is simple, A$^2$ = A. By induction, A$^n$ = 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=Π A$_i$ is semisimple. If (a$_i$) is an element of Rad(A) and e$_1$ is the multiplicative identity in A$_1$ (all simple algebras possess a multiplicative identity), then (a$_1$, a$_2$, ...) · (e$_1$, 0, ...) = (a$_1$, 0..., 0) lies in some nilpotent ideal of Π A$_i$. This implies, for all b in A$_1$, a$_1$b is nilpotent in A$_1$, i.e. a$_1$ ∈ Rad(A$_1$). So a$_1$ = 0. Similarly, a$_i$ = 0 for all other i.
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.

## Characterization

Let A be a finite-dimensional semisimple algebra, and
    $\{0\}=J_{0}\subset \cdots \subset J_{n}\subset A$
be a composition series of A, then A is isomorphic to the following Cartesian product:
    $A\simeq J_{1}\times J_{2}/J_{1}\times J_{3}/J_{2}\times ...\times J_{n}/J_{n-1}\times A/J_{n}$
where each
    $J_{i+1}/J_{i}\,$
is a simple algebra.
The proof can be sketched as follows. First, invoking the assumption that A is semisimple, one can show that the J$_1$ is a simple algebra (therefore unital). So J$_1$ is a unital subalgebra and an ideal of J$_2$. Therefore, one can decompose
    $J_{2}\simeq J_{1}\times J_{2}/J_{1}.$
By maximality of J$_1$ as an ideal in J$_2$ and also the semisimplicity of A, the algebra
    $J_{2}/J_{1}\,$
is simple. Proceed by induction in similar fashion proves the claim. For example, J$_3$ is the Cartesian product of simple algebras
    $J_{3}\simeq J_{2}\times J_{3}/J_{2}\simeq J_{1}\times J_{2}/J_{1}\times J_{3}/J_{2}.$
The above result can be restated in a different way. For a semisimple algebra A = A$_1$ ×...× A$_n$ expressed in terms of its simple factors, consider the units e$_i$ ∈ A$_i$. The elements E$_i$ = (0,...,e$_i$,...,0) are idempotent elements in A and they lie in the center of A. Furthermore, E$_i$ A = A$_i$, E$_i$E$_j$ = 0 for i ≠ j, and Σ E$_i$ = 1, the multiplicative identity in A.
Therefore, for every semisimple algebra A, there exists idempotents {E$_i$} in the center of A, such that
* E$_i$E$_j$ = 0 for i ≠ j (such a set of idempotents is called central orthogonal),
* Σ E$_i$ = 1,
* A is isomorphic to the Cartesian product of simple algebras E$_1$ A ×...× E$_n$ A.

## Classification

A theorem due to Joseph Wedderburn completely classifies finite-dimensional semisimple algebras over a field $k$. Any such algebra is isomorphic to a finite product $\prod M_{n_{i}}(D_{i})$ where the $n_{i}$ are natural numbers, the $D_{i}$ are division algebras over $k$, and $M_{n_{i}}(D_{i})$ is the algebra of $n_{i}\times n_{i}$ matrices over $D_{i}$. This product is unique up to permutation of the factors.
This theorem was later generalized by Emil Artin to semisimple rings. This more general result is called the Wedderburn–Artin theorem.