Hidden Matrices

We investigate subrings of an n × n matrix ring which, despite appearing otherwise, are themselves full rings of n × n matrices; that is, are hidden matrices . In general, this problem is subtle, but we give fairly complete results in a number of situations. For example, we prove: Theorem A. Let K be an ideal of a ring R and suppose that T = (R ij ) is a tiled subring of M n (R) containing M n (K). Suppose that R ii = R jj for all i and j and that R ii /K ≅ M n (D), for some ring D. Then T ≅ M n (S), for a ring S that we describe explicitly. The subtleties are illustrated by the following theorem: Theorem B. Let H denote the ring of integer quaternions and let p be an odd prime number. Set R = H + M 2 (pH), where H is identified with the ring of scalar matrices inside M 2 (H). Then R≅M 2 (S), for some ring S, if and only if p ≡1 (mod 4).