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)^[1] 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.^[2]

Theorem[]

Let $R$ be a (Artinian) semisimple ring. Then $R$ 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 $i$ .

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.

Corollary 1[]

The Wedderburn–Artin theorem implies that every simple ring that is finite-dimensional over a division ring is isomorphic to an n-by-n matrix ring over a division ring D, where both n and D are uniquely determined.^[2] This is Joseph Wedderburn's original result. Emil Artin later generalized it to the case of left or right Artinian rings. In particular, if $k$ is an algebraically closed field, then the matrix ring having entries from $k$ is the only finite dimensional Artinian simple algebra over $k$ .

Corollary 2[]

Let $k$ be an algebraically closed field. Let $R$ be a semisimple ring, that is a finite-dimensional $k$ -algebra. 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 $k$ .

Consequence[]

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 $K$ .

References[]

^ Semisimple rings are necessarily Artinian rings. Some authors use "semisimple" to mean the ring has a trivial Jacobson radical. For Artinian rings, the two notions are equivalent, so "Artinian" is included here to eliminate that ambiguity.
^ ^a ^b John A. Beachy (1999). Introductory Lectures on Rings and Modules. Cambridge University Press. p. 156. ISBN 978-0-521-64407-5.

P. M. Cohn (2003) Basic Algebra: Groups, Rings, and Fields, pages 137–9.
J.H.M. Wedderburn (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". 5: 251–260. Cite journal requires |journal= (help)

[1] Semisimple rings are necessarily Artinian rings. Some authors use "semisimple" to mean the ring has a trivial Jacobson radical. For Artinian rings, the two notions are equivalent, so "Artinian" is included here to eliminate that ambiguity.

[Beachy1999-2] John A. Beachy (1999). Introductory Lectures on Rings and Modules. Cambridge University Press. p. 156. ISBN 978-0-521-64407-5.

[1]

[2]