JOURNAL OF ALGEBRA 101, 365402 (1986) Completely O-Simple Semigroups of Quotients* JOHNFOUNTAIN AND MARIO PETRICH Department of Mathematics, University of York, Heslington, York YOl.500, England Communicated by P. M. Cohn Received October 3, 1984 1. INTRODUCTION Many definitions of semigroups of quotients have been proposed and studied. For a survey, the reader may consult Weinert’s paper [ 131. In this paper we introduce yet another kind of semigroup of quotients. The motivation for our definition comes, to some extent, from examining a con- nection between semigroups and rings. In the introduction to Chapter 3 of [ 11, Clifford and Preston outline a method of obtaining a completely O-simple semigroup from an Artinian simple ring. Given such a ring Q, the identity can be written as a sum l=e,+ ... + e, of pair-wise orthogonal primitive idempotents. Then putting Q, = eiQei we get a subsemigroup & = U {e,: 1 < i, j < n} of Q and Q is completely O-simple. Up to isomorphism, Q is independent of the choice of the idempotents e1 ,..., e, and is isomorphic to A0 (n, A*, n, I) where A is the division ring Qll, n = {l,..., n}, and I is the n x n identity matrix. The Artin-Wedderburn theorem tells us that Q is isomorphic to the matrix ring M,(A). It follows from this theorem that Q possesses a set of matrix units, that is, a subset {eV: i, j= l,..., n} whose elements satisfy eiiehk= eik if j=h =o if j#h. The elements eii, i= l,..., n, form a set of pair-wise orthogonal primitive idempotents and putting Q, = ei,Qejj, we have Q, = e,D where D consists of all elements of Q which commute with all the eg. The set D is a division subring of Q and is isomorphic to A. * The work reported in this paper was carried out at the Institut fiir Mathematik, Univer- sitiit Wien, while the second author was a visiting professor there. The tirst author was sup- ported by a European Science Exchange Programme award from the Royal Society. 365 0021-8693/86 $3.00 48l/lOl/2-6 Copyright rc 1986 by Academx Press, Inc. All rights of reproductmn m any form reserved.