A Theorem of Burnside on Matrix Rings

invariant subspaces in C2, and in fact, r, s, rs = (? l), together with the identity matrix clearly form a basis of M2(C). Burnside's Theorem (and its subsequent generalization by Frobenius and Schur in [5]) proved to be a fundamental result in the representation theory of groups, and has appeared in many books on that subject. From a ring-theoretic perspective, [2] and [5] yield a more general result, nowadays also called Burnside's Theorem, which can be formulated as follows.