Artin-Wedderburn theorem

In mathematics, the Artin-Wedderburn theorem is a classification theorem for simple rings. It applies in particular to the case of rings that are an algebra over a field, and finite-dimensional over it (Wedderburn's original case).

The theorem states that a simple ring R that is Artinian is isomorphic with the nxn matrix ring over a division ring D, for some integer n. In that case the center of D must be a field K. Therefore R is a K-algebra, and itself has K as center: R is a central simple algebra over K.

When K is the real number field R, possible examples are the nxn matrix rings over R, the complex numbers C, and the quaternion ring H only. When K is C, just nxn complex matrices.