HIDDEN MATRICES

We investigate subrings of an n x n matrix ring which, despite appeari ng otherwise, are themselves full rings of n x n matrices; that is, ar e hidden matrices. In general, this problem is subtle, but we give fai rly 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 congruent to M(n)(D), for som e ring D. Then T congruent to M(n)(S), for a ring S that we describe e xplicitly. The subtleties are illustrated by the following theorem: TH EOREM 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 congruent to M(2)(S), for some ring S, if and only if p drop 1 (mod 4).