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

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