Σ-uniserial Modules and Their Properties

The close association of abelian group theory and the theory of modules have been extensively studied in the literatures. In fact, the theory of abelian groups is one of the principal motives of new research in module theory. As it is well-known, module theory can only be processed by generalizing the theory of abelian groups that provide novel viewpoints of various structures for torsion abelian groups. The theory of torsion abelian groups is significant as it generates the natural problems in QT AG-module theory. The notion of QT AG (torsion abelian group like) module is one of the most important tools in module theory. Its importance lies behind the fact that this module can be applied in order to generalized torsion abelian group accurately. Significant work on QT AG-module was produced by many authors, concentrating on establishing when torsion abelian groups are actually QT AG-modules. There are two rather natural problems which arise in connection with the Σ-uniserial modules. Namely: The QT AG-module M is Σ-uniserial if and only if all N-high submodules of M are Σ-uniserial, for some basic submodules N of M, and M is not a Σ-uniserial module if and only if it contains a proper (ω + 1)-projective submodule. The current work explores these two problems for QT AG-modules. Some related concepts and problems are also considered. Our global aim here is to review the relationship between the aspects of group theory in the form of torsion abelian groups and theory of modules in the form of QT AG-modules.