ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring R is a complete n x n matrix ring, so R congruent to M-n(S) for some ring S, if and only if it contains a set of n x n matrix units {e(ij)}(i,)(n) (j=1). A more recent and less known result states that a ring R is a complete (m + n) x (m + n) matrix ring if and only if, R contains three elements, a, b, and f, satisfying the two relations af(m)+f(n)b = 1 and f(m+n) = 0. In many instances the two elements a and b can be replaced by appropriate powers a(i) and a(j) of a single element a respectively. In general very little is known about the structure of the ring S. In this article we study in depth the case m = n = 1 when R congruent to M-2(S). More specifically we study the universal algebra over a commutative ring A with elements x and y that satisfy the relations x(i)y + yx(j) = 1 and y(2) = 0. We describe completely the structure of these A-algebras and their underlying rings when gcd(i,j) = 1. Finally we obtain results that fully determine when there are surjections onto M-2(F) when F is a base field Q or Z(p) for a prime number p.