Some remarks on \(\Phi\)-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 \(\Phi\)-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 \(\Phi:\mathfrak{T}(M)\longrightarrow M_{P}\) 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.