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 = ≠ . As the terminology suggests, simple algebras are semisimple. The only possible ideals of a simple algebra ''A'' are ''A'' and . 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, ''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 is a finite collection of simple algebras, then their Cartesian product A=Π ''Ai'' is semisimple. If (''ai'') 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 ''Ai''. 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, ''ai'' = 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 : 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 ×...× ''An'' expressed in terms of its simple factors, consider the units ''ei'' ∈ ''Ai''. The elements ''Ei'' = (0,...,''ei'',...,0) are idempotent elements in ''A'' and they lie in the center of ''A''. Furthermore, ''Ei A'' = ''Ai'', ''EiEj'' = 0 for ''i'' ≠ ''j'', and Σ ''Ei'' = 1, the multiplicative identity in ''A''. Therefore, for every semisimple algebra ''A'', there exists idempotents in the center of ''A'', such that #''EiEj'' = 0 for ''i'' ≠ ''j'' (such a set of idempotents is called '' central orthogonal''), #Σ ''Ei'' = 1, #''A'' is isomorphic to the Cartesian product of simple algebras ''E''1 ''A'' ×...× ''En 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. This theorem was later generalized byReferences