On \(\Phi\)-Dedekind, \(\Phi\)-Prüfer and \(\Phi\)-Bezout modules
Let \(R\) be a commutative ring with identity. In this paper the authors generalize the notions of Dedekind, Prüfer and Bézout modules to the context of modules in the class \(\mathbb{H}=\{M\,|\,M\text{ is an \(R\)-module and }\mathrm{Nil}(M)\text{ is a divided prime submodule of }M\}\). Let \(M\) be an \(R\)-module in \(\mathbb{H}\). Then the module \(\mathrm{Nil}(M)\) is a prime submodule of M. Set \(T=(R\setminus Z(R))\cap (R\setminus Z(M))\), \(\mathfrak{T}(M) = T^{-1}M\) and \(P := (\mathrm{Nil}(M) :_R M)\) prime ideal of \(R\). The mapping \(\Phi : \mathfrak T(M)\to M_P\), defined by \(\Phi(x/s) = x/s\) is an \(R\)-module homomorphism whose restriction to M is also an \(R\)-module homomorphism given by \(\Phi(x) = x/1\) for every \(x\in M\). A submodule \(N\) of \(M\) such that \(N\nsubseteq \mathrm{Nil}(M)\) (i.e., \(N\) is a nonnil submodule of \(M\)) is said to be \(\Phi\)-invertible if \(\Phi(N)\) is an invertible submodule of \(\Phi(M)\). The module \(M\) is said to be a \(\Phi\)-Prüfer module if every finitely generated nonnil submodule of \(M\) is \(\Phi\)-invertible and a \(\Phi\)-Dedekind module if every nonnil submodule of \(M\) is \(\Phi\)-invertible. Moreover, \(M\) is said to be a \(\Phi\)-Bézout module if \(\Phi(N)\) is a principal ideal of \(\Phi(M)\) for every finitely generated submodule \(N\) of the \(R\)-module \(M\). Some properties of these families of modules and their relation with the notions of Prüfer, Dedekind and Bézout modules are discussed.
scientific article; zbMATH DE number 7188077