In algebra, the Wedderburn–Artin 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.
Theorem
Let be a (Artinian) semisimple ring. Then the Wedderburn–Artin theorem states that is isomorphic to the product of finitely many -by- matrix rings <math>M_{n_i}(D_i)</math> 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 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 <math>R</math> is semisimple. Then the right <math>R</math>-module <math>R_R</math> is isomorphic to a finite direct sum of simple modules (which are the same as minimal right ideals of <math>R</math>). Write this direct sum as
: <math>
R_R \;\cong\; \bigoplus_{i=1}^m I_i^{\oplus n_i}
</math>
where the <math>I_i</math> are mutually nonisomorphic simple right <math>R</math>-modules, the th one appearing with multiplicity <math>n_i</math>. This gives an isomorphism of endomorphism rings
: <math>
\mathrm{End}(R_R)
\;\cong\;
\bigoplus_{i=1}^m \mathrm{End}\big(I_i^{\oplus n_i}\big)
</math>
and we can identify <math>\mathrm{End}\big(I_i^{\oplus n_i}\big)</math> with a ring of matrices
: <math>
\mathrm{End}\big(I_i^{\oplus n_i}\big) \;\cong\; M_{n_i}\big(\mathrm{End}(I_i)\big)
</math>
where the endomorphism ring <math>\mathrm{End}(I_i)</math> of <math>I_i</math> is a division ring by Schur's lemma, because <math>I_i</math> is simple. Since <math>R \cong \mathrm{End}(R_R)</math> we conclude
: <math>
R \;\cong\;
\bigoplus_{i=1}^m M_{n_i}\big(\mathrm{End}(I_i)\big)
\,.
</math>
Here we used right modules because <math>R \cong \mathrm{End}(R_R)</math>; if we used left modules <math>R</math> would be isomorphic to the opposite algebra of <math>\mathrm{End}({}_R R)</math>, 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 <math> k </math>, 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 <math> k </math>. Then is a finite product <math>\textstyle \prod_{i=1}^r M_{n_i}(k) </math> where the <math> n_i </math> are positive integers and <math> M_{n_i}(k) </math> is the algebra of <math> n_i \times n_i </math> matrices over <math> k </math>.
Furthermore, the Wedderburn–Artin theorem reduces the problem of classifying finite-dimensional central simple algebras over a field <math> k </math> to the problem of classifying finite-dimensional central division algebras over <math> k </math>: that is, division algebras over <math> k </math> whose center is <math> k </math>. It implies that any finite-dimensional central simple algebra over <math> k </math> is isomorphic to a matrix algebra <math>\textstyle M_{n}(D) </math>
where <math>D</math> is a finite-dimensional central division algebra over <math> k </math>.
See also
- Maschke's theorem
- Brauer group
- Jacobson density theorem
- Hypercomplex number
- Emil Artin
- Joseph Wedderburn