Wedderburn–Artin theorem
In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that a 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.
Theorem
Let be a semisimple ring. Then the Wedderburn–Artin 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 Wedderburn–Artin 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 field extension|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.
Proof
There are various proofs of the Wedderburn–Artin 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. 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 Artinian ring#Simple [Artinian ring|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, 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 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 Wedderburn–Artin 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.