Artin–Wedderburn theorem

In algebra, the Artin–Wedderburn 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 ni-by-ni matrix rings over division rings Di, for some integers ni, 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 semisimple ring. Then R is isomorphic to a product of finitely many ni-by-ni matrix rings ${\displaystyle M_{n_{i}}(D_{i})}$  over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i.

If R is a finite-dimensional semisimple k-algebra, then each Di in the above statement is a finite-dimensional division algebra over k. The center of each Di 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 Artin–Wedderburn 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 ${\displaystyle k}$  is an algebraically closed field, then the matrix ring having entries from ${\displaystyle k}$  is the only finite dimensional division algebra over ${\displaystyle k}$ .

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 ${\displaystyle \textstyle \prod _{i=1}^{r}M_{n_{i}}(k)}$  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 k.

Consequence

The Artin–Wedderburn 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

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