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 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-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 A be a semisimple R-algebra.

(i) A\cong M_{n_1}(D_1)\oplus\dots\oplus M_{n_r}(D_r) for some natural numbers n_1,\dots,n_r and R-division algebras D_1,\dots,D_r.

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

(iii) Conversely, suppose A=M_{n_1}(D_1)\oplus\dots\oplus M_{n_r}(D_r), then A is a right (and left) semisimple R-algebra.

Some Notes

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

Example: R=\mathbb{R} and A=M_2(\mathbb{R})\oplus M_2(\mathbb{R}). Then A is semisimple by Wedderburn’s theorem.

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