On Some Classes of Modules with Generalized Extending Condition

Abstract

In this work, we present purely generalized extending (PGCS) modules. A module D is called PGCS if, for each submodule E of D, there exists a pure submodule X of D containing E for which fracXE is singular. This idea extends purely (generalized) extending modules. Moreover, PGCS modules are closed under direct summands. Furthermore, we establish the strongly purely generalized extending modules (SPGCS), where the pure submodule X is required to be fully invariant. This new structure lies properly between strongly extending modules and PGCS modules. We investigate many algebraic properties of these modules and characterize their relationships with other well-known classes of modules.