Quasi-injective $S$-systems and their $S$-endomorphism semigroup

Abstract

Patterned after the theory of modules over a ring, P, BERTHIAUME [1] introduced the concepts of injective and weakly-injective 5-systems. He exhibited examples of such 5-systems and showed that properties holding true for a right ring module need not hold for a right S-system. For example, a weakly injective S-system need not be injective; in ring theory, this is part of Baer's Theorem. In this paper, we study another weak form of injectivity called quasi-injectivity. Quasi-injective modules have been studied by JOHNSON and WONG Recently M. SATYANARAYANA Our paper is a study of quasi-injective S-systems and their S-endomorphism semigroup. We characterize the smallest quasi-injective essential extension of an S-system M s contained in /(M^), its injective hull. Further we give conditions for Hom^ (M, M) to be (VON NUEMAN) regular and obtain as corollaries a result of M. BOTERO DE MEZA [2] deahng with the regularity of the maximal right quotient semigroup (S) of a semigroup S, and a generalization for S-systems of Faith and Utumi's result on the regularity of the endomorphism ring of a quasiinjective module.