On annihilator multiplication modules
Suat Koç · Journal of Algebra and Its Applications · 2026
An [Formula: see text]-module [Formula: see text] is said to be an annihilator multiplication module if for each [Formula: see text], there exists a finitely generated ideal [Formula: see text] of [Formula: see text] such that [Formula: see text]. 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 paper 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 non-torsion annihilator multiplication modules, the equality [Formula: see text] 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.