Quasi-injective $S$-systems and their $S$-endomorphism semigroup
Jr. Lopez Antonio M., John K. Luedeman · Czechoslovak Mathematical Journal · 1979
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 [6], FAITH and UTUMI [3], and B. OSOFSKY [8], among others.Recently M. SATYANARAYANA [9] investigated quasi-and weaklyinjective S-systems.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 ß(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.1. PRELIMINARIES