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

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 p of R with (0 : R M) ⊆ p, there exists a w-submodule N p of M such that (M/N p ) w is w-cofinitely generated and (M[p]) w ⊆ N p ⊆ (0 : M p), where M[p] = s∈R\p s(0 : M p). Besides, we show that the w-operations are semi-star operations rather than star operations in general.