A Cohen-type theorem for w-Noetherian modules. Issue 11 (19th August 2022)

• Record Type: Journal Article
• Authors: Huang, Ke; Wang, Fanggui
• Is Part Of: Communications in algebra. Volume 50:Issue 11(2022)
• Page Start: 4882
• Page End: 4890
• Publication Date: 2022-08-19
• Subjects: Cohen's theorem -- distinguished prime w-submodules -- w-Noetherian modules -- prime w-ideals -- prime w-submodules -- w-finite type; 13A15 -- 13E99 -- 13F99; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2022.2077951
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: An expanded distinguished prime w -submodule of a w -module M is defined by M w e ( P ) = { x ∈ M | s x ∈ ( P M ) w for some s ∈ R ∖ P } where P is a prime w -ideal of R . Using this, we prove a Cohen-type theorem for w -Noetherian modules: A w -finite type R -module M is a w -Noetherian module if and only if for every prime w -ideal P of R with Ann ( M ) ⊆ P, there exists a w -finite type w -submodule N of M such that ( P M ) w ⊆ N ⊆ M w e ( P ) . As byproducts, among others, we get several conditions such that PM is a w -module where P is a prime w -ideal of R and R -module M is a w -module.