On morphic modules over commutative rings (Q6547691)

This paper focuses on extending the concept of \textit{morphic rings} to modules over commutative rings. The authors investigate properties of morphic modules by comparing them with related structures such as von Neumann regular modules and Bézout modules.\N\NThe core definition presented is that an \(R\)-module \(M\) is morphic if, for each \(m \in M\), there exists \(a \in R\) such that:\N\[\NRm = \operatorname{ann}_M(a) \quad \text{and} \quad \operatorname{ann}_R(m) = Ra + \operatorname{ann}_R(M).\N\]\NThe paper establishes the equivalence of morphic rings and morphic \(R\)-modules. Several examples and propositions illustrate the theory, including the results on the intersection of cyclic submodules, the characterization of Baer submodules, and the stability of morphic modules under various contexts like homomorphisms and localization.\N\NThe authors also connect morphic modules with von Neumann regular modules and Bézout modules, showing that a finitely generated von Neumann regular module is also a morphic module. Additionally, they present several classes of modules, including torsion-free and reduced modules, and study how these relate to the morphic property.\N\NThe paper concludes with an open question regarding the connections between the newly defined morphic modules and previously studied notions in the literature, particularly those by Nicholson and Sánchez Campos.