On annihilator multiplication modules

An A-module E is said to be an annihilator multiplication module if for each e ∈ 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 Ass 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.