  Let R be a commutative noetherian ring. Denote by mod R the category of
finitely generated R-modules. In the present paper, we first provide various
sufficient (and necessary) conditions for a full subcategory of mod R to be a
Serre subcategory, which include several refinements of theorems of Stanley and
Wang and of Takahashi with simpler proofs. Next we consider when an IKE-closed
subcategory of mod R is a torsionfree class. We investigate certain modules out
of which all modules of finite length can be built by taking direct summands
and extensions, and then we apply it to show that the IKE-closed subcategories
of mod R are torsionfree classes in the case where R is a certain numerical
semigroup ring.
