Some remarks on Φ-sharp modules

Abstract

The purpose of this paper is to introduce some new classes of modules which is closely related to the classes of sharp modules, pseudo-Dedekind modules and T V -modules. In this

paper we introduce the concepts of Φ-sharp modules, Φ-pseudo- Dedekind modules and Φ-T V -modules. Let R be a commutative

ring with identity and set H = {M | M is an R-module and Nil(M) is a divided prime submodule of M}. For an R-module M ∈ H, set T = (R \ Z(M)) ∩ (R \ Z(R)), T(M) = T −1 (M) and P := (Nil(M) :R M). In this case the mapping Φ : T(M) −→ MP given by Φ(x/s) = x/s is an R-module homomorphism. The restriction of Φ to M is also an R-module homomorphism from M in to MP given by Φ(m/1) = m/1 for every m ∈ M. An R-module M ∈ H is called a Φ-sharp module if for every nonnil submodules N, L of M and every nonnil ideal I of R with N ⊇ IL, there exist a nonnil ideal I ′ ⊇ I of R and a submodule L

′ ⊇ L of M such that N = I ′L ′ . We prove that Many of the properties and characterizations of sharp modules may be extended to Φ-sharp modules, but some can not.