2019/07/11 by Agnarsson, Geir, Mendelson, Samuel S.
#15B33 #16S15 #16S50 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1907.05335
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 \eij\i,j=1n. 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 afm+fnb=1 and fm+n=0. In many instances the two elements a and b can be replaced by appropriate powers ai and aj 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≅ M2(S). More specifically we study the universal algebra over a commutative ring A with elements x and y that satisfy the relations xiy+yxj=1 and y2=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 M2(\mathbb F) when \mathbb F is a base field \mathbb Q or \mathbb Zp for a prime number p.