  Let $R$ be a commutative noetherian ring and denote by $\mathsf{mod} R$ the
category of finitely generated $R$-modules. In this paper, we study KE-closed
subcategories of $\mathsf{mod} R$, that is, additive subcategories closed under
kernels and extensions. We first give a characterization of KE-closed
subcategories: a KE-closed subcategory is a torsion-free class in a
torsion-free class. As an immediate application of the dual statement, we give
a conceptual proof of Stanley-Wang's result about narrow subcategories. Next,
we classify the KE-closed subcategories of $\mathsf{mod} R$ when $\mathrm{dim}
R \le 1$ and when $R$ is a two-dimensional normal domain. More precisely, in
the former case, we prove that KE-closed subcategories coincide with
torsion-free classes in $\mathsf{mod} R$. Moreover, this condition implies
$\mathrm{dim} R \le 1$ when $R$ is a homomorphic image of a Cohen-Macaulay ring
(e.g. a finitely generated algebra over a regular ring). Thus, we give a
complete answer for the title.
