Corollary 3

Let K be a commutative field, and let A be a K-algebra of finite degree with simple underlying ring. There exist an integer $r$ and a K-algebra D of finite degree over K that is a field such that A is isomorphic to $M_r(D)$. In particular, if K is algebraically closed, then A is isomorphic to a matrix algebra over K.

Let S be a simple left A-module; it is a finite-dimensional K-vector space over K. Its commutant is therefore an algebra of finite degree over K. The first assertion then follows from Lemma 1. If K is algebraically closed, then D = K by Theorem 1 of VIII, p. 47.