  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.
