Let R be a commutative Noetherian ring, I an ideal of R and M an arbitrary R-module. Let S be a Serre subcategory of the category of R-modules. It is shown that the R-module Exti R(R=I;M) belongs to S, for all i 0, if and only if the R-module Exti R(R=I;M) belongs to S, for all 0 i ara(I). As an immediate consequence, we prove that if R is a Noetherian (resp. (R;m) is a Noetherian local) ring of dimension d, then the R-module Exti R(R=I;M) belongs to S, for all i 0 if and only if the R-module Exti R(R=I;M) belongs to S, for all 0 i d+1 (resp. for all 0 i d). Also it is shown that if I is a principal ideal up to radical, then the category of I-cominimax (resp. I-weakly co nite) modules is an Abelian full subcategory of the category of R-modules.
