On Annihilator Multiplication Modules

Abstract

An $A$-module $E$ is said to be an \textit{annihilator multiplication module} if for each $e\in E$, there exists a finitely generated ideal $I$ of $A$ such that $ann(e)=ann(IE)$. This class of modules is quite large, as it contains multiplication modules, von Neumann regular modules, finitely generated Baer modules, torsion-free modules, and simple modules. This article provides a comprehensive investigation into the algebraic properties of annihilator multiplication modules, and establishes new characterizations for several important classes of rings/modules, including multiplication modules, torsion-free modules, simple modules, uniserial modules, injective modules and Noetherian von Neumann regular rings. Furthermore, we present a construction method using trivial extensions to produce annihilator multiplication rings that are not multiplication rings. In addition, we prove that, for such modules, the equality $As{s}_A(E)=Ass(A)$ holds, thereby providing a precise connection between module-theoretic and ring-theoretic prime structures. Finally, we provide various examples to demonstrate the above equality may fail if the condition of being annihilator multiplication module is omitted.

This version includes minor revisions and improved proofs