EN
Let R be a commutative noetherian ring, let 𝔞 be an ideal of R, and let 𝓢 be a subcategory of the category of R-modules. The condition $C_{𝔞}$, defined for R-modules, was introduced by Aghapournahr and Melkersson (2008) in order to study when the local cohomology modules relative to 𝔞 belong to 𝓢. In this paper, we define and study the class $𝓢_{𝔞}$ consisting of all modules satisfying $C_{𝔞}$. If 𝔞 and 𝔟 are ideals of R, we get a necessary and sufficient condition for 𝓢 to satisfy $C_{𝔞}$ and $C_{𝔟}$ simultaneously. We also find some sufficient conditions under which 𝓢 satisfies $C_{𝔞}$. As an application, we investigate when local cohomology modules lie in a Serre subcategory.