Wedderburn–Artin theorem

Template:Short description In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian)Template:Efn semisimple ring R is isomorphic to a product of finitely many Template:Math-by-Template:Math matrix rings over division rings Template:Math, for some integers Template:Math, both of which are uniquely determined up to permutation of the index Template:Mvar. 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.Template:Sfn

Theorem

Let Template:Mvar be a (Artinian) semisimple ring. Then the Wedderburn–Artin theorem states that Template:Mvar is isomorphic to a product of finitely many Template:Math-by-Template:Math matrix rings $M_{n_i}(D_i)$ over division rings Template:Math, for some integers Template:Math, both of which are uniquely determined up to permutation of the index Template:Mvar.

There is also a version of the Wedderburn–Artin theorem for algebras over a field Template:Mvar. If Template:Mvar is a finite-dimensional semisimple Template:Mvar-algebra, then each Template:Math in the above statement is a finite-dimensional division algebra over Template:Mvar. The center of each Template:Math need not be Template:Mvar; it could be a finite extension of Template:Mvar.

Note that if Template:Mvar 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.Template:SfnTemplate:Sfn A common modern oneTemplate:Sfn 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

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

where the $I_i$ are mutually nonisomorphic simple right $R$-modules, the Template:Mvarth one appearing with multiplicity $n_i$. This gives an isomorphism of endomorphism rings

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

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

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

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

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

Here we used right modules because $R\cong \mathrm{E}\mathrm{n}\mathrm{d}(R_R)$; if we used left modules $R$ would be isomorphic to the opposite algebra of $\mathrm{E}\mathrm{n}\mathrm{d}({}_{R}R)$, 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 $k$, where both n and D are uniquely determined.Template:Sfn 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 Template:Mvar be a semisimple ring that is a finite-dimensional algebra over an algebraically closed field $k$. Then Template:Mvar is a finite product ${\prod }_{i=1}^r{M}_{n_i}(k)$ 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 $M_n(D)$ where $D$ is a finite-dimensional central division algebra over $k$.

See also

• Maschke's theorem
• Brauer group
• Jacobson density theorem
• Hypercomplex number
• Emil Artin
• Joseph Wedderburn

References

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