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 J1 is a simple algebra (therefore unital). So J1 is a unital subalgebra and an ideal of J2. 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. 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 {Ei} 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 E1 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 by Emil Artin to semisimple rings. This more general result is called the Artin-Wedderburn theorem. References Springer Encyclopedia of Mathematics Algebras