On Cohen’s theorem for modules

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

“…In 1994, Smith [10] extended Cohen's Theorem from rings to modules, that is, a finitely generated R-module M is Noetherian if and only if the submodules pM of M are finitely generated for every prime ideal p of R, if and only if M(p) is finitely generated for each prime ideal p of R with (0 : R M) ⊆ p, where M(p) = {x ∈ M | sx ∈ pM for some s ∈ R \ p}. In 2021, Parkash and Kour [8] generalized the Smith's result on Noetherian modules and obtained that a finitely generated R-module M is Noetherian if and only if for every prime ideal p of R with (0 : R M) ⊆ p, there exists a finitely generated submodule N p of M such that pM ⊆ N p ⊆ M(p). Recently, the author et al [13] gave a w-analogue of Parkash and Kour's result which states that a GV-torsion-free w-finite type R-module M is w-Noetherian if and only if for every prime w-ideal p of R with (0 : R M) ⊆ p, there exists a w-finite type submodule N p of M such that pM ⊆ N p ⊆ M(p).…”

A note on the Cohen type theorem and the Eakin-Nagata type theorem for uniformly $S$-Noetherian rings

“…Recently, Parkash and Kour[7] generalized and extended Cohen type theorem to Noetherian modules: a finitely generated R-module M is Noetherian if and only if for every prime ideal p of R with Ann(M) ⊆ p, there exists a finitely generated submodule N p of M such that pM ⊆ N p ⊆ M(p), where M(p) := {x ∈ M |…”

On two versions of Cohen's theorem for modules

“…[11, Theorem 2.1] Let R be a ring and M a finitely generated Rmodule. Then M is Noetherian if and only if for every prime ideal p of R with Ann(M) ⊆ p, there exists a finitely generated submodule N p of M such that pM ⊆ N p ⊆ M(p).…”