As a direct corollary, the Artin-Wedderburn theorem implies that every simple ring that is finite-dimensional over a division ring (a simple algebra) is a matrix ring. This is Joseph Wedderburn's original result. Emil Artin later generalized it to the case of Artinian rings.

Note that if R is a finite-dimensional simple algebra over a division ring E, that D can be strictly larger than E. For example, matrix rings over the complex numbers are finite-dimensional simple algebras over the real numbers.

The Artin-Wedderburn theorem reduces classifying simple rings over a division ring to classifying division rings that contain a given division ring. This in turn can be simplified: The center of D must be a field K. Therefore R is a K-algebra, and itself has K as its center. A finite-dimensional simple algebra R is thus a central simple algebra over K. Thus the Artin-Wedderburn theorem reduces the problem of classifying finite-dimensional central simple algebras to the problem of classifying division rings with given center.

Examples

Let R by the field of real numbers, C be the field of complex numbers, and H the quaternions.