Finitely generated modules over a P.I.D. and the Smith Normal Form

I’m teaching Graduate Algebra this semester, and I wanted to record here the proof I gave in class of the (existence part of the) structure theorem for finitely generated modules over a PID. It’s a standard argument, based on the existence of the Smith Normal Form for a matrix with entries in a PID, but it’s surprisingly hard to find a concise and accessible reference.

Henry John Stephen Smith (1826-1883)

We assume familiarity with basic definitions in the theory of modules over a (commutative) ring. Our goal is to prove the following:

Theorem: Let R be a principal ideal domain and let M be a finitely generated R-module. Then M is isomorphic to a direct sum of cyclic R-modules. More precisely, there are a non-negative integer r (called the rank of M) and elements d_1,\ldots,d_n \in M (called the invariant factors of M) such that d_i \mid d_{i+1} for all i=1,\ldots,n-1 and M \cong R^r \oplus R/(d_1) \oplus R/(d_2) \oplus \cdots \oplus R/(d_n).

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