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

TL;DR

This paper characterizes $w$-Artinian modules via $w$-cofinitely generated submodules and prime $w$-ideals, and discusses the nature of $w$-operations as semi-star rather than star operations.

Contribution

It provides a Cohen-type characterization for $w$-Artinian modules and clarifies the properties of $w$-operations as semi-star operations.

Findings

$w$-Artinian modules are characterized by $w$-cofinitely generated conditions.

The existence of specific $w$-submodules related to prime $w$-ideals is established.

$w$-operations are shown to be semi-star, not star, operations.

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 $\mathfrak{p}$ of $R$ with $(0:_RM)\subseteq \mathfrak{p}$, there exists a $w$-submodule $N^\mathfrak{p}$ of $M$ such that $(M/N^\mathfrak{p})_w$ is $w$-cofinitely generated and $(M[\mathfrak{p}])_w\subseteq N^\mathfrak{p} \subseteq (0:_M\mathfrak{p})$, where $M[\mathfrak{p}]=\bigcap\limits_{s\in R \setminus \mathfrak{p}}s(0:_M\mathfrak{p}).$ Besides, we show that the $w$-operations are semi-star operations rather than star operations in general.