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 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.

Theorem

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 $M_{n_{i}}(D_{i})$ 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.

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 ith 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 $k$.

See also

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

References

• Beachy, John A. (1999). Introductory Lectures on Rings and Modules. Cambridge University Press. p. 156. ISBN 978-0-521-64407-5.
• Cohn, P. M. (2003). Basic Algebra: Groups, Rings, and Fields. pp. 137–139.
• Henderson, D.W. (1965). "A short proof of Wedderburn's theorem". The American Mathematical Monthly. 72 (4): 385–386. doi:10.2307/2313499. JSTOR 2313499.
• Nicholson, William K. (1993). "A short proof of the Wedderburn–Artin theorem" (PDF). New Zealand J. Math. 22: 83–86.
• Wedderburn, J.H.M. (1908). "On Hypercomplex Numbers". Proceedings of the London Mathematical Society. 6: 77–118. doi:10.1112/plms/s2-6.1.77.
• Artin, E. (1927). "Zur Theorie der hyperkomplexen Zahlen". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 5: 251–260. doi:10.1007/BF02952526. JFM 53.0114.03.