Wedderburn–Artin theorem

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 $$M_{n_i}(D_i)$$ 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

R_R \;\cong\; \bigoplus_{i=1}^m I_i^{\oplus n_i} $$ where the $$I_i$$ are mutually nonisomorphic simple right $$R$$-modules, the $i$th one appearing with multiplicity $$n_i$$. This gives an isomorphism of endomorphism rings

\mathrm{End}(R_R) \;\cong\; \bigoplus_{i=1}^m \mathrm{End}\big(I_i^{\oplus n_i}\big) $$ and we can identify $$\mathrm{End}\big(I_i^{\oplus n_i}\big)$$ with a ring of matrices

\mathrm{End}\big(I_i^{\oplus n_i}\big) \;\cong\; M_{n_i}\big(\mathrm{End}(I_i)\big) $$ where the endomorphism ring $$\mathrm{End}(I_i)$$ of $$I_i$$ is a division ring by Schur's lemma, because $$I_i$$ is simple. Since $$R \cong \mathrm{End}(R_R)$$ we conclude

R \;\cong\; \bigoplus_{i=1}^m M_{n_i}\big(\mathrm{End}(I_i)\big) \,. $$

Here we used right modules because $$R \cong \mathrm{End}(R_R)$$; if we used left modules $$R$$ would be isomorphic to the opposite algebra of $$\mathrm{End}({}_R R)$$, 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 $$\textstyle \prod_{i=1}^r M_{n_i}(k) $$ where the $$ n_i $$ are positive integers and $$ M_{n_i}(k) $$ is the algebra of $$ n_i \times n_i $$ 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 $$\textstyle M_{n}(D) $$ where $$D$$ is a finite-dimensional central division algebra over $$ k $$.