When Is a Subcategory Serre or Torsion-Free?

Abstract

Let $R$ be a commutative noetherian ring. Denote by $\mathrm{mod}R$ the category of finitely generated $R$-modules. In the present paper, we first provide various sufficient (and necessary) conditions for a full subcategory of $\mathrm{mod}R$ 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 $\mathrm{mod}R$ is a torsion-free 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 $\mathrm{mod}R$ are torsion-free classes in the case where $R$ is a certain numerical semigroup ring.