On the Relation β_ss^* in Module Theory

Abstract

In this study, some features of the Goldie ss-lifting modules concept and generalizations of this module class are given with the help of the relation  defined in the article (Gömleksiz & Nişancı Türkmen, 2023). In this relation , which is defined as submodules  and  of the module  provided that   and      is determined by conditions. The important properties of this relation in Goldie-ss-lifting modules have been studied. In this article, Goldie-ss-supplemented modules are considered as a special case of Goldie*-supplemented modules and Goldie-ss-lifting modules are considered as a special case of Goldie*-lifting modules, and the basic module structure theorems are included with the help of the relation  , which is more special than the relation . The classification of the ss-semi-local modules was made using the relation . It has been proved that the factor modules of Goldie-ss-supplemented modules are also Goldie-ss-supplemented modules. It has been shown that , which is the set of equivalence classes according to the relation  for submodules of a module  , has a monoid structure. With the help of fully invariant submodules, it has been shown that every direct summand of a Goldie-ss-supplemented module is a Goldie-ss-supplemented module. In addition, ss-supplemented modules and Goldie-ss-supplemented modules classes, and ss-lifting modules and Goldie-ss-lifting modules classes were compared.