Artin–Wedderburn Theorem

A classification theorem stating that any semisimple Artinian ring (equivalently any finite-dimensional semisimple algebra over a field) is isomorphic to a finite direct product of matrix algebras over division rings; in particular, simple artinian rings are matrix algebras over division rings.

The Artin–Wedderburn theorem classifies semisimple Artinian rings: any semisimple Artinian ring is isomorphic to a finite direct product of matrix algebras over division rings. In particular, a semisimple finite‑dimensional algebra over a field is a finite direct sum of full matrix algebras over division algebras.