Classifying subcategories of modules over a PID

Abstract

Let $R$ be a commutative ring. A full additive subcategory $\C$ of $R$-modules is triangulated if whenever two terms of a short exact sequence belong to $\C$, then so does the third term. In this note we give a classification of triangulated subcategories of finitely generated modules over a principal ideal domain. As a corollary we show that in the category of finitely generated modules over a PID, thick subcategories (triangulated subcategories closed under direct summands), wide subcategories (abelian subcategories closed under extensions) and Serre subcategories (wide subcategories closed under kernels) coincide and correspond to specialisation closed subsets of $\Spec(R)$.