On 2-Nil Primary Submodules of Modules Over Commutative Rings

Abstract

We introduce and examine the class of 2-nil primary submodules for modules over commutative rings. This class is obtained by combining the nilpotency option occurring in 2-nil submodules with the radical condition that appears in primary-type submodule theory. Several equivalent descriptions are obtained; these descriptions are expressed by residual submodules, by ideal products, and by products of submodules. In the setting of multiplication modules, we compare the new class with primary submodules and connect it with 2-nil primary ideals for faithful finitely generated multiplication modules. We then discuss how the property is transferred through module homomorphisms, quotient modules, intersections, localization, idealization, direct products, and amalgamated algebra constructions. The examples clarify that many of the implications established here are not reversible without extra assumptions. The paper therefore gives a systematic module-theoretic treatment of a condition governed simultaneously by nilpotent and radical alternatives.