In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring R is isomorphic to a product of finitely many ni -by- ni matrix rings over division rings Di , for some integers ni , both of which are uniquely determined up to permutation of the index i. In particular, any simple left or right Artinian ring is isomorphic to an n-by-n matrix ring over a division ring D, where both n and D are uniquely determined.

Theorem

Let R be a (Artinian) semisimple ring. Then the Wedderburn–Artin theorem states that R is isomorphic to a product of finitely many ni -by- ni matrix rings over division rings Di , for some integers ni , both of which are uniquely determined up to permutation of the index i. There is also a version of the Wedderburn–Artin theorem for algebras over a field k. If R is a finite-dimensional semisimple k-algebra, then each Di in the above statement is a finite-dimensional division algebra over k. The center of each Di need not be k; it could be a finite extension of k. Note that if R is a finite-dimensional simple algebra over a division ring E, D need not be contained in E. For example, matrix rings over the complex numbers are finite-dimensional simple algebras over the real numbers.

Proof

There are various proofs of the Wedderburn–Artin theorem. A common modern one takes the following approach. Suppose the ring R is semisimple. Then the right R-module R_R is isomorphic to a finite direct sum of simple modules (which are the same as minimal right ideals of R). Write this direct sum as where the I_i are mutually nonisomorphic simple right R-modules, the ith one appearing with multiplicity n_i. This gives an isomorphism of endomorphism rings and we can identify with a ring of matrices where the endomorphism ring of I_i is a division ring by Schur's lemma, because I_i is simple. Since we conclude Here we used right modules because ; if we used left modules R would be isomorphic to the opposite algebra of, but the proof would still go through. To see this proof in a larger context, see Decomposition of a module. For the proof of an important special case, see Simple Artinian ring.

Consequences

Since a finite-dimensional algebra over a field is Artinian, the Wedderburn–Artin theorem implies that every finite-dimensional simple algebra over a field is isomorphic to an n-by-n matrix ring over some finite-dimensional division algebra D over k, where both n and D are uniquely determined. This was shown by Joseph Wedderburn. Emil Artin later generalized this result to the case of simple left or right Artinian rings. Since the only finite-dimensional division algebra over an algebraically closed field is the field itself, the Wedderburn–Artin theorem has strong consequences in this case. Let R be a semisimple ring that is a finite-dimensional algebra over an algebraically closed field k. Then R is a finite product where the n_i are positive integers and M_{n_i}(k) is the algebra of matrices over k. Furthermore, the Wedderburn–Artin theorem reduces the problem of classifying finite-dimensional central simple algebras over a field k to the problem of classifying finite-dimensional central division algebras over k: that is, division algebras over k whose center is k. It implies that any finite-dimensional central simple algebra over k is isomorphic to a matrix algebra where D is a finite-dimensional central division algebra over k.

Citations