Wedderburn–Artin theorem

Proof

There are various proofs of the Wedderburn–Artin theorem. A common modern one takes the following approach.

Suppose the ring ${\displaystyle R}$ is semisimple. Then the right ${\displaystyle R}$-module ${\displaystyle R_R}$ is isomorphic to a finite direct sum of simple modules (which are the same as minimal right ideals of ${\displaystyle R}$). Write this direct sum as

${\displaystyle R_R\cong \underset{i=1}{\overset{m}{⨁}}{I}_i^{\oplus n_i}}$

where the ${\displaystyle I_i}$ are mutually nonisomorphic simple right ${\displaystyle R}$-modules, the ith one appearing with multiplicity ${\displaystyle n_i}$. This gives an isomorphism of endomorphism rings

${\displaystyle \mathrm{E}\mathrm{n}\mathrm{d}(R_R)\cong \underset{i=1}{\overset{m}{⨁}}\mathrm{E}\mathrm{n}\mathrm{d}(I_i^{\oplus n_i})}$

and we can identify ${\displaystyle \mathrm{E}\mathrm{n}\mathrm{d}(I_i^{\oplus n_i})}$ with a ring of matrices

${\displaystyle \mathrm{E}\mathrm{n}\mathrm{d}(I_i^{\oplus n_i})\cong M_{n_i}(\mathrm{E}\mathrm{n}\mathrm{d}(I_i))}$

where the endomorphism ring ${\displaystyle \mathrm{E}\mathrm{n}\mathrm{d}(I_i)}$ of ${\displaystyle I_i}$ is a division ring by Schur's lemma, because ${\displaystyle I_i}$ is simple. Since ${\displaystyle R\cong \mathrm{E}\mathrm{n}\mathrm{d}(R_R)}$ we conclude

${\displaystyle R\cong \underset{i=1}{\overset{m}{⨁}}{M}_{n_i}(\mathrm{E}\mathrm{n}\mathrm{d}(I_i)).}$

Here we used right modules because ${\displaystyle R\cong \mathrm{E}\mathrm{n}\mathrm{d}(R_R)}$; if we used left modules ${\displaystyle R}$ would be isomorphic to the opposite algebra of ${\displaystyle \mathrm{E}\mathrm{n}\mathrm{d}({}_{R}R)}$, 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 ${\displaystyle k}$, 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 ${\displaystyle k}$. Then R is a finite product ${\displaystyle \prod _{i=1}^r{M}_{n_i}(k)}$ where the ${\displaystyle n_i}$ are positive integers and ${\displaystyle M_{n_i}(k)}$ is the algebra of ${\displaystyle n_i\times n_i}$ matrices over ${\displaystyle k}$.

Furthermore, the Wedderburn–Artin theorem reduces the problem of classifying finite-dimensional central simple algebras over a field ${\displaystyle k}$ to the problem of classifying finite-dimensional central division algebras over ${\displaystyle k}$: that is, division algebras over ${\displaystyle k}$ whose center is ${\displaystyle k}$. It implies that any finite-dimensional central simple algebra over ${\displaystyle k}$ is isomorphic to a matrix algebra ${\displaystyle M_n(D)}$ where ${\displaystyle D}$ is a finite-dimensional central division algebra over ${\displaystyle k}$.