A note on a Cohen-type theorem for w-Artinian modules

2023/01/07 by Xiaolei Zhang, Zhang, Xiaolei
Mathematics · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2301.02772

openalex publication_date 2023/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we prove that a w-module M is w-Artinian if and only if it is w-cofinitely generated and for every prime w-ideal \mathfrakp of R with (0:_RM)⊆ \mathfrakp, there exists a w-submodule N^\mathfrakp of M such that (M/N^\mathfrakp)_w is w-cofinitely generated and (M[\mathfrakp])_w⊆ N^\mathfrakp ⊆ (0:_M\mathfrakp), where M[\mathfrakp]=\bigcap_s∈ R ∖ \mathfrakps(0:_M\mathfrakp). Besides, we show that the w-operations are semi-star operations rather than star operations in general.

Related

• On Cohen's theorem for Artinian modules
• Cosupport of artinian modules
• On two versions of Cohen's theorem for modules
• Hopfian and co-Hopfian modules over Artinian rings
• Some results on local cohomology modules