On j-Artinian Modules Over Commutative Rings
Dilara Erdemir, Najib Mahdou, El Houssaine Oubouhou, Ünsal Tekir
TL;DR
The paper generalizes Artinian module theory by introducing $\jmath$-Artinian $R$-modules, where every descending chain of $\jmath$-submodules stabilizes. It provides equivalent characterizations of the $\jmath$-Artinian property via descending chains, minimal elements, and finite intersections, and proves an Akizuki-type theorem: every cyclic $\jmath$-Artinian module is $\jmath$-Noetherian. It also analyzes the relation between $M$ and its quotient $M/\jmath$, shows that quotient modules inherit Artinian properties, and extends these results to exact sequences, localization, and amalgamated structures. The results support extending classical Artinian/Noetherian theory to $\jmath$-relative settings and pave the way for applications to amalgamation and pullback constructions in module theory.
Abstract
Researchers introduced the notion of j-Artinian rings in [3] and obtained significant results concerning this new class of rings. Motivated by their definition and findings, we extend the study to modules by introducing the concept of j-Artinian modules. Recall from [9] that, if R is a commutative ring with identity, M is an R-module, and j is a submodule of M, then a submodule N of M is called a j-submodule if N \not\subseteq j. We say that M is a j-Artinian R-module if every descending chain of j-submodules becomes stationary. In this paper, we provide a characterization of j-Artinian modules. Moreover, we establish an analogue of Akizuki's theorem in this context and discuss its extension to amalgamated structures.