A definition for weakly finite $R$-modules is as follow:
Definition: Let ($R$,$m$) a local ring. Let $S$ be the largest class of $R$-modules satisfying the following four properties:
(1) If $M \in S$, then Hom($\frac{R}{m}, M$) is finitely generated.
(2) If $M$ is a non-zero element of $S$ and $r$ is an regular element, then $\frac{M}{rM} \in S$ is non-zero and dim($\frac{M}{rM}$) = dim$M$ - 1.
(3) If $M \in S$, then #Ass($M$) is finite.
(4) If $M \in S$, then $\frac{M}{\Gamma_m(M)} \in S$.
We say that an $R$-module $M$ is weakly finite, if $M \in S$.
I would like to know if the set of weakly finite R-modules is Serre subcategory of the category of R-modules.
