P-Polyform Modules

Abstract

An R-module M is called polyform if every essential submodule of M is rational. The main goal of this paper is to give a new class of modules named P-polyform modules. This class of modules are contained properly in the class of polyform modules. Several properties of this concept are introduced and compared with those which is known in the concept of polyform modules. Another characterization of the definition of P-polyform modules is given as an analogue to that in the concept of polyform modules. So we proved that a module M is P-polyform if and only if HomR(NK, M)= 0, for each essential submodule N of M, which is pure in M, and a submodule K of M, with N⊆K⊆M. The relationships between this class of modules and some other related concepts are discussed such as monoform, QI-monoform, monoform, essentially quasi-Dedekind, essentially prime, purely quasi-Dedekind, ESQD and St-polyform modules. Furthermore, purely St-polyform is defined and its relationship with the P-polyform module is studied.