On Almost Locally \(h\)-pure \(QTAG\)-Modules

Abstract

Following [5], A module MR is called a QTAG-module if every finitely generated submodule of every homomorphic image of MR is a direct sum of uniserial modules. This is a very fascinating structure and many mathematicians worked to generalize the results for abelian groups for these modules. Many interesting results have been surfaced, but there is a lot to explore.
The purpose of this paper is to introduce and investigate the concept of almost locally  \(h\)-pure QTAG-modules. A QTAG-module M is almost locally  \(h\)-pure if every finitely generated submodule of M may be embedded in a finitely generated \(h\)-pure submodule of M. It was found that a QTAG-module is almost locally \(h\)-pure if and only if it is h-reduced, a direct sum of almost locally h-pure submodules is almost locally h-pure and every submodule of an almost locally \(h\)-pure QTAG-module is almost locally  \(h\)-pure.