A Cohen-type theorem for w-Noetherian modules. Issue 11 (19th August 2022) - Huang, Ke; Wang, Fanggui
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.