Simple-direct-modules

Abstract

A right R-module M is called simple-direct-injective if, whenever, A and B are simple submodules of M with AB, and (BM)-M-circle plus, then A(circle plus)M. Dually, M is called simple-direct-projective if, whenever, A and B are submodules of M with M/AB(circle plus)M and B simple, then A(circle plus)M. In this paper, we continue our investigation of these classes of modules strengthening many of the established results on the subject. For example, we show that a ring R is uniserial (artinian serial) with J(2)(R)=0 iff every simple-direct-projective right R-module is an SSP-module (SIP-module) iff every simple-direct-injective right R-module is an SIP-module (SSP-module).