When is a subcategory Serre or torsionfree?

TL;DR

This paper characterizes when subcategories of finitely generated modules over a noetherian ring are Serre or torsionfree, providing new criteria and applying them to specific algebraic structures like numerical semigroup rings.

Contribution

It offers new necessary and sufficient conditions for subcategories to be Serre or torsionfree, refining previous theorems with simpler proofs and applying results to numerical semigroup rings.

Findings

Provided criteria for Serre subcategories in mod R.

Characterized IKE-closed subcategories as torsionfree in certain rings.

Extended understanding of subcategory structures in module categories.

Abstract

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.