Some remarks on $\Phi$-sharp modules

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 $\Phi$-sharp modules, $\Phi$-pseudo-Dedekind modules and $\Phi$-$TV$-modules. Let $R$ be a commutative ring with identity and set $\mathbb{H}=\lbrace M\mid M$ is an $R$-module and $\operatorname{Nil}(M)$ is a divided prime submodule of $M\rbrace$. For an $R$-module $M\in\mathbb{H}$, set $T=(R\setminus Z(M))\cap (R\setminus Z(R))$, $\mathfrak{T}(M)=T^{-1}(M)$ and $P:=(\operatorname{Nil}(M):_{R}M)$. In this case the mapping $\Phi\colon\mathfrak{T}(M)\longrightarrow M_{P}$ given by $\Phi(x/s)=x/s$ is an $R$-module homomorphism. The restriction of $\Phi$ to $M$ is also an $R$-module homomorphism from $M$ in to $M_{P}$ given by $\Phi(m/1)=m/1$ for every $m\in M$. An $R$-module $M\in \mathbb{H}$ is called a $\Phi$-sharp module if for every nonnil submodules $N,L$ of $M$ and every nonnil ideal $I$ of $R$ with $N\supseteq IL$, there exist a nonnil ideal $I'\supseteq I$ of $R$ and a submodule $L'\supseteq L$ of $M$ such that $N=I'L'$. We prove that Many of the properties and characterizations of sharp modules may be extended to $\Phi$-sharp modules, but some can not.