Matrix algebras (or matrix rings) may be defined over any ring S. We shall regard such a matrix algebra as an S-algebra. Let us denote the complete algebra of n x n matrices over S by M_n(S). It will often be convenient to regard M_n(S) as the endomorphism ring of the free S-module S^((n)). We will then speak of S^((n)) as the natural S-module associated with M_n(S).