The Stacks project

Proof. A PID is a Noetherian Bézout ring. By Lemma 15.124.8 it suffices to prove the result if $M$ is torsion. Since $M$ is finite, this means that the annihilator of $M$ is nonzero. Say $fM = 0$ for some $f \in R$ nonzero. Then we can think of $M$ as a module over $R/fR$. Since $R/fR$ is Noetherian of dimension $0$ (small detail omitted) we see that $R/fR = \prod R_ j$ is a finite product of Artinian local rings $R_ i$ (Algebra, Proposition 10.60.7). Each $R_ i$, being a local ring and a quotient of a PID, is a generalized valuation ring in the sense of Lemma 15.124.2 (small detail omitted). Write $M = \prod M_ j$ with $M_ j = e_ j M$ where $e_ j \in R/fR$ is the idempotent corresponding to the factor $R_ j$. By Lemma 15.124.3 we see that $M_ j = \bigoplus _{i = 1, \ldots , n_ j} R_ j/\overline{f}_{ji}R_ j$ for some $\overline{f}_{ji} \in R_ j$. Choose lifts $f_{ji} \in R$ and choose $g_{ji} \in R$ with $(g_{ji}) = (f_ j, f_{ji})$. Then we conclude that

as an $R$-module which finishes the proof. $\square$