On a class of presentations of matrix algebras

Abstract

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 ∼ = M n (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 Z p for a prime number p.2010 MSC: 15B33, 16S15, 16S50.

On a class of presentations of matrix algebras

“…Since by Lemma 2.5 x is invertible, then so is b in L, and we then obtain from the above two equations that f (i + j − 1) = b j−1 and f (i + j) = 0. We therefore get that In [Agn96], it is shown that for A = k a field, R(k; i, j, 1, 1) always maps to some M N (k) and…”

“…The purpose of this article is in part to extend the results in [Agn96] that partly answer the above Question 1.7. The article is organized as follows.…”

“…This gives an explicit isomorphism of M 2 (Q(λ)) into a subring of M 2 (M 3 (Q)) = M 6 (Q). This means that the smallest N in [Agn96] for which there is a non-trivial Q-algebra homomorphism R(Q; 4, 3, 1, 1) → M N (Q) is here N = 6. We will see that this observation agrees with Theorem 3.8 in the following Section 3.…”

“…By Theorem 1.2 we have R(A; i, j, m, n) ∼ = M m+n (S) for some A-algebra S. As we have very little information about S, we introduce the following sets similarly as in [Agn96].…”

On a Special Presentation of Matrix Algebras

“…It is therefore natural to ask what happens if the first relation in Theorem 1.2 is replaced by a i f m + f n a j = 1. The ring R is by Theorem 1.2 a complete (m + n) × (m + n) matrix ring, but it could be the trivial ring; in [1] it is shown that if a ring R contains elements a and b such that ab m + b n a = 1 and b m+n = 0 where m = n, then R is the trivial ring. This result together with Theorem 1.3 strongly suggest the study of the universal ring that contains two elements a and f that satisfy a i f m + f n a j = 1 and f m+n = 0.…”