Wedderburn’s Structure Theorem

In abstract 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 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-nmatrix ring over a division ring D, where both n and D are uniquely determined. (Wikipedia)

This is quite a powerful theorem, as it allows semisimple rings/algebras to be “represented” by a finite direct sum of matrix rings over division rings. This is in the spirit of Representation theory, which tries to convert algebraic objects into objects in linear algebra, which is relatively well understood.

Wedderburn’s Structure Theorem

Let be a semisimple -algebra.

(i) for some natural numbers and -division algebras .

(ii) The pair is unique up to isomorphism and order of arrangement.

(iii) Conversely, suppose , then is a right (and left) semisimple -algebra.

Some Notes

The definition of a semisimple -algebra is: An -algebra is semisimple if is semisimple as a right -module.

Example: and . Then is semisimple by Wedderburn’s theorem.