Fully Polyform Modules and Related Concepts

An R-module M is called a polyform module if every essential submodule of M is rational. The main objective of this paper is to introduce a new concept of modules named fully polyform modules. This kind of module is contained in the class of polyform modules. We study in detail fully polyform modules, so several properties of this concept are investigated. Other characterizations and partial characterisations (i.e., satisfied by certain conditions) of the definition of fully polyform module analogous to those known in the concept of a polyform module are given and discussed. For instance, we proved that a module M is a fully polyform module if and only if M)=0 for each P-essential submodule N of M and for each V≤M with NÍVÍM. Relationships between this class of modules and some other related concepts are discussed such as monoform, QI-monoform, essentially quasi-Dedekind, essentially prime and St-polyform modules. Moreover, useful concepts and their influence or relationships with fully polyform modules such as P-uniform and Pe-prime modules are introduced.