Melkersson condition for extension of Serre subcategories

Let R be a commutative noetherian ring and let $$\mathfrak {a}$$ an ideal of R. In this paper, we study certain condition, namely $$C_{\mathfrak {a}}$$ , introduced by Aghapournahr Melkersson, on the extension two subcategories R-modules. We develop some main results Yoshizawa (Proc Am Math Soc 140(7):2293–2305, 2012; J Commut Algebra 13:137–155, 2021). As example subcategories, weakly Laskerian modules find conditions under which local cohomology module lie in arbitrary Serre subcategory. Eventually, investigate cofiniteness modules.