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 TV-modules. In this paper we introduce the concepts of Φ-sharp modules, Φ-pseudo-Dedekind modules and Φ-TV-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):RM). 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.