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×n matrix ring, so R ∼ = Mn(S) for some ring S, if and only if it contains a set of n × n matrix units {e ij } n i,j=1. A more recent and less known result states that a ring R is a complete (m + n) × (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 ∼ = 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 Zp for a prime number p.

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