The Stacks project

Lemma 15.121.9. Let $R$ be a PID. Every finite $R$-module $M$ is of isomorphic to a module of the form

\[ R^{\oplus r} \oplus \bigoplus \nolimits _{i = 1, \ldots , n} R/f_ iR \]

for some $r, n \geq 0$ and $f_1, \ldots , f_ n \in R$ nonzero.

Proof. A PID is a Noetherian B├ęzout ring. By Lemma 15.121.8 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 15.121.2 (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 15.121.3 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

\[ M \cong \bigoplus R/g_{ji}R \]

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


Comments (2)

Comment #6339 by comment_bot on

Although this is pretty trivial, it may be worthwhile saying in the statement that, in particular, every finitely generated projective module over a PID is free.

Comment #6440 by on

Of course you are right and maybe we should add this. Inside the Stacks project, I think I would use this lemma directly for that result if it was ever needed. Anyway, I am happy for somebody else to edit the latex according to what you said and submit it to stacks.project@gmail.com.


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