In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras.
The theorem states that an (Artinian)[a] 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.
[1] 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.
There are various proofs of the Wedderburn–Artin theorem.
[2][3] A common modern one[4] takes the following approach.
Suppose the ring
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 ith one appearing with multiplicity
This gives an isomorphism of endomorphism rings and we can identify
is a division ring by Schur's lemma, because
would be isomorphic to the opposite algebra of
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 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
[1] 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
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
It implies that any finite-dimensional central simple algebra over
is isomorphic to a matrix algebra
is a finite-dimensional central division algebra over