In algebra, the WedderburnâÂÂArtin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that a(n Artinian) semisimple ring R is isomorphic to the product of finitely many -by- matrix rings over division rings , for some integers , both of which are uniquely determined up to permutation of the index . 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.
Let be a (Artinian) semisimple ring. Then the WedderburnâÂÂArtin theorem states that is isomorphic to the product of finitely many -by- matrix rings over division rings , for some integers , both of which are uniquely determined up to permutation of the index .
There is also a version of the WedderburnâÂÂArtin theorem for algebras over a field . If is a finite-dimensional semisimple -algebra, then each in the above statement is a finite-dimensional division algebra over . The center of each need not be ; it could be a finite extension of .
Note that if 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.
There are various proofs of the WedderburnâÂÂArtin theorem. A common modern one takes the following approach.
Suppose the ring is semisimple. Then the right -module is isomorphic to a finite direct sum of simple modules (which are the same as minimal right ideals of ). Write this direct sum as
where the are mutually nonisomorphic simple right -modules, the th one appearing with multiplicity . This gives an isomorphism of endomorphism rings
and we can identify with a ring of matrices
where the endomorphism ring of is a division ring by Schur's lemma, because is simple. Since we conclude
Here we used right modules because ; if we used left modules 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.
Since a finite-dimensional algebra over a field is Artinian, the WedderburnâÂÂArtin 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 , 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 WedderburnâÂÂArtin theorem has strong consequences in this case. Let be a semisimple ring that is a finite-dimensional algebra over an algebraically closed field . Then is a finite product where the are positive integers and is the algebra of matrices over .
Furthermore, the WedderburnâÂÂArtin theorem reduces the problem of classifying finite-dimensional central simple algebras over a field to the problem of classifying finite-dimensional central division algebras over : that is, division algebras over whose center is . It implies that any finite-dimensional central simple algebra over is isomorphic to a matrix algebra where is a finite-dimensional central division algebra over .