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)^[a] semisimple ring R is isomorphic to a product of finitely many n_i-by-n_i matrix rings over division rings D_i, for some integers n_i, 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 n_i-by-n_i matrix rings ${\displaystyle M_{n_{i}}(D_{i})}$ over division rings D_i, for some integers n_i, 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 D_i in the above statement is a finite-dimensional division algebra over k. The center of each D_i 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 ${\displaystyle R}$ 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

${\displaystyle R_{R}\;\cong \;\bigoplus _{i=1}^{m}I_{i}^{\oplus n_{i}}}$

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

${\displaystyle \mathrm {End} (R_{R})\;\cong \;\bigoplus _{i=1}^{m}\mathrm {End} {\big (}I_{i}^{\oplus n_{i}}{\big )}}$

and we can identify ${\displaystyle \mathrm {End} {\big (}I_{i}^{\oplus n_{i}}{\big )}}$ with a ring of matrices

${\displaystyle \mathrm {End} {\big (}I_{i}^{\oplus n_{i}}{\big )}\;\cong \;M_{n_{i}}{\big (}\mathrm {End} (I_{i}){\big )}}$

where the endomorphism ring ${\displaystyle \mathrm {End} (I_{i})}$ of ${\displaystyle I_{i}}$ is a division ring by Schur's lemma, because ${\displaystyle I_{i}}$ is simple. Since ${\displaystyle R\cong \mathrm {End} (R_{R})}$ we conclude

${\displaystyle R\;\cong \;\bigoplus _{i=1}^{m}M_{n_{i}}{\big (}\mathrm {End} (I_{i}){\big )}\,.}$

Here we used right modules because ${\displaystyle R\cong \mathrm {End} (R_{R})}$; if we used left modules ${\displaystyle R}$ would be isomorphic to the opposite algebra of ${\displaystyle \mathrm {End} ({}_{R}R)}$, 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.

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.^[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 ${\displaystyle k}$. Then R is a finite product ${\displaystyle \textstyle \prod _{i=1}^{r}M_{n_{i}}(k)}$ 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 ${\displaystyle \textstyle M_{n}(D)}$ where ${\displaystyle D}$ is a finite-dimensional central division algebra over ${\displaystyle k}$.

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