A simple matrix semigroup which is not completely simple

Abstract

Let K be a field, M(n,K) be the multiplicatlve semlgroup of all n• matrices over K, GL(n,K) be the group of all invertible matrices over K. If A~ M(n,K) then by LA we denote a linear envelope of A. The set A is said to be irreducible if LA=M(n,K). In ~I] the following problem was posed (~9,problem ~): does there exist a O-simple irreducible semigroup in M(n,K) which is not completely O-simple; does there exist a simple irreducible subsemigroup of GL(a,K) which is not a group? In [2S the importance of this problem for the theory of semigroup rings was noted. The purpose of the present note is to answer both questions affirmatively. To this end we construct an example of a simple semlgroup S in GL(n,K) which is not a group. Then the semlgroup S o with adjoined zero is O-simple and is not completel~ O-simple. THEOreM. Let Q be the ~ of rational numbers. is ~ SimDle ~ ~ S of GL(2,Q) ~I f~ is not a ~ . P r o o f. Let G be a subgroup generated in GL(2,Q)