On Annihilator Multiplication Modules
Suat Koç
TL;DR
The paper introduces annihilator multiplication modules (AM modules) over commutative rings with 1 and situates them among multiplication, Baer, and vn-regular modules to bridge module- and ring-theoretic prime structures. It develops essential properties, stability results, and connections to classical classes (multiplication, vn-regular, Baer, torsion-free, and simple modules), culminating in the key equality $Ass_{A}(E)=Ass(A)$ for AM modules. Through these, it derives characterizations of torsion-free modules, multiplication modules, injective modules, and principal ideal von Neumann regular rings, linking module annihilators to ring primes. The framework provides a unified approach to understanding how module-theoretic prime structures reflect and determine ring-theoretic prime structures, with potential applications in identifying vn-regular and Baer-like rings via annihilator behavior.
Abstract
An $A$-module $E$ is an annihilator multiplication module if, for each $e\in E$, there is a finitely generated ideal $I$ of $A$ such that $ann(e)=ann(IE)$. In this paper, we investigate fundamental properties of annihilator multiplication modules and employ them as a framework for characterizing significant classes of rings and modules, including torsion-free modules, multiplication modules, injective modules, and principal ideal von Neumann regular rings. In addition, we establish that, for such modules, the equality $Ass_{A}(E)=Ass(A)$ holds, thereby providing a precise connection between module-theoretic and ring-theoretic prime structures.