IE-closed subcategories of commutative rings are torsion-free classes

Abstract

Let C be a subcategory of the category of finitely generated R-modules over a commutative noetheian ring R. We prove that, if C is closed under images and extensions (which we call an IE-closed subcategory), then C is closed under submodules, and hence is a torsion-free class. This result complements Stanley--Wang's result in some sense and, furthermore, provides a complete answer to the question posed by Iima--Matsui--Shimada--Takahashi. The proof relies on the general theory of IE-closed subcategories in an abelian category, which states that IE-closed subcategories are precisely the intersections of torsion classes and torsion-free classes. Additionally, we completely characterize right noetherian rings such that every IE-closed subcategory (or torsion-free class) is a Serre subcategory.