Extended S-Pure Regular Modules

Abstract

    In previous studies, es-pure submodule and es-pure flat module were presented, when S semiradical property and this is what led us to introduce es-pure regular module. In this paper, the concept of es-regular modules have introduced as a significant area of study. We say that M is extended S- pure regular (or, es-regular for short) module if for every submodule of M is es-pure submodule of M. By presenting several examples, we illustrate the nuances of es-regularity in different algebraic contexts. This paper investigates the properties and implications of es-regular modules. We begin by establishing foundational definitions, we characterize es-regular modules and exploring the relationship between es-regular modules and other well-known classes of modules, such as projective and semisimple modules, establish connections to semisimple rings and explore links with flatness. Also, we discussed the relationship between es-pure regular module and es-pure regular ring.