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]

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

Proof

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.

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.[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 ${\displaystyle k}$.

See also

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

Notes

1. By the definition used here, semisimple rings are automatically Artinian rings. However, some authors use "semisimple" differently, to mean that the ring has a trivial Jacobson radical. For Artinian rings, the two notions are equivalent, so "Artinian" is included here to eliminate that ambiguity.

Citations

1. Beachy 1999
2. Henderson 1965
3. Nicholson 1993
4. Cohn 2003

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.