The endomorphism structure of simple faithful S-acts

Abstract

In ring theory Schur’s Lemma says that, given a primitive ring R with a simple faithful R-module M, the set of all endomorphisms of M has the structure of a skew eld over which M can be considered as a vector space. This provides the vector space used in Jacobson’s classical density theorem that characterizes primitive rings as dense rings of linear mappings. Since for the study of primitive rings End M is of such an importance, it seems natural to investigate End M in the semigroup case as well. Oehmke showed in [2] that a primitive commutative semigroup is a cyclic group of prime order. We will give a dierent proof of this property and will further show how it is connected with determining the structure of End M. Denition 1. Let S be a semigroup and M a left S-act. For an element m 2 M we dene the annihilator of m as Ann m := f(a; b) 2 S S j am = bmg