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.

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

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Theorem:Let be a principal ideal domain and let be a finitely generated -module. Then is isomorphic to a direct sum of cyclic -modules. More precisely, there are a non-negative integer (called therankof ) and elements (called theinvariant factorsof ) such that for all and .