On nonnil-coherent modules and nonnil-Noetherian modules (Q2111843)

This paper is concerned with a generalization of coherent modules and Noetherian modules. Let \(R\) be a commutative ring with prime nilradical. Then \(R\) is called a \(\phi\)-ring if \(\mathrm{Nil}(R)\) is divided prime, that is \(\mathrm{Nil}(R)\subseteq xR\) for all \(x\in R\setminus \mathrm{Nil}(R)\).\par Let \(R\) be a \(\phi\)-ring and \(M\) be an \(R\)-module. A submodule \(N\) of \(M\) is said to be a \(\phi\)-submodule of \(M\) if \(M/N\) is a \(\phi\)-torsion module, that is for every \(x\in M\) there exists \(s\in R\setminus \mathrm{Nil}(R)\) such that \(sx\in N\). An \(R\)-module \(M\) is said to be nonnil-coherent if \(M\) is finitely generated and every finitely generated \(\phi\)-submodule of \(M\) is finitely presented. \par Let \(R\) be a \(\phi\)-ring. An \(R\)-module \(M\) is said to be nonnil-Noetherian if every \(\phi\)-submodule of \(M\) is finitely generated. \par The paper start with sevral characterizations of nonnil-coherent rings, and then some properties that characterize nonnil-coherent modules. Also the authors gave some properties that characterize nonnil-Noetherian modules.\par Among other results, the authors study the transfer of the properties of nonnil-coherent rings and nonnil-Noetherian rings in trivial ring extensions. The paper also include the study of the transfer of the properties of being \(\phi\)-coherent rings and nonnil-Noetherian rings in an amalgamation algebra along an ideal.