Pure C 3 modules

Abstract

Within the paper, we study the class of modules in which the direct sum of two disjoint summands that is a pure submodule is also a summand. This class of modules is considered PC3 modules, a generalization of C3 modules. In addition to pure-injective and PC2 modules that belong to the class of PC3 modules, which also include the semisimple, continuous, indecomposable, and regular modules. We gave new characterizations of many rings in respect of PC3 modules, namely semisimple rings, pure-semisimple rings, von Neumann regular rings, Noetherian rings, pure-hereditary rings, pure-V-rings, etc. Moreover, we also discuss the PC3 envelope and PC3 cover of a module and introduce the pure continuous modules as the generalization of the continuous modules and also introduce the pure quasi-continuous modules as the generalization of the quasi-continuous modules. © 2023 World Scientific Publishing Company.