# Semisimple algebra

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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 not nilpotent, then *A* is semisimple. 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*is semisimple. If (

_{i}*a*) is an element of Rad(

_{i}*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 ∏

*A*. This implies, for all

_{i}*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*= 0 for all other

_{i}*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. The following is a semisimple algebra that appears not to be of this form. Let *A* be an algebra with Rad(*A*) ≠ *A*. The quotient algebra *B* = *A* ⁄ Rad(*A*) is semisimple: If *J* is a nonzero nilpotent ideal in *B*, then its preimage under the natural projection map is a nilpotent ideal in *A* which is strictly larger than Rad(*A*), a contradiction.

## Characterization

Let *A* be a finite-dimensional semisimple algebra, and

be a composition series of *A*, then *A* is isomorphic to the following Cartesian product:

where each

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

By maximality of *J*_{1} as an ideal in *J*_{2} and also the semisimplicity of *A*, the algebra

is simple. Proceed by induction in similar fashion proves the claim. For example, *J*_{3} is the Cartesian product of simple algebras

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*. The elements

_{i}*E*= (0,...,

_{i}*e*,...,0) are idempotent elements in

_{i}*A*and they lie in the center of

*A*. Furthermore,

*E*=

_{i}A*A*,

_{i}*E*= 0 for

_{i}E_{j}*i*≠

*j*, and Σ

*E*= 1, the multiplicative identity in

_{i}*A*.

Therefore, for every semisimple algebra *A*, there exists idempotents {*E _{i}*} in the center of

*A*, such that

*E*= 0 for_{i}E_{j}*i*≠*j*(such a set of idempotents is called*central orthogonal*),- Σ
*E*= 1,_{i} *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 . Any such algebra is isomorphic to a finite product where the are natural numbers, the are division algebras over , and is the algebra of matrices over . This product is unique up to permutation of the factors.^{[1]}

This theorem was later generalized by Emil Artin to semisimple rings. This more general result is called the Artin-Wedderburn theorem.

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}