Some remarks on \(\Phi\)-sharp modules

Ahmad Yousefian Darani, Mahdi Rahmatinia


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.


\(\Phi\)-sharp module, \(\Phi\)-pseudo-Dedekind module, \(\Phi\)-Dedekind module, \(\Phi\)-\(TV\) module

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