November | 2022 | Matt Baker's Math Blog

In my previous post, I presented a proof of the existence portion of the structure theorem for finitely generated modules over a PID based on the Smith Normal Form of a matrix. In this post, I’d like to explain how the uniqueness portion of that theorem is actually a special case of a more general result, called Fitting’s Lemma, which holds for arbitrary commutative rings.

We begin by proving that one can characterize the diagonal entries in the Smith Normal Form of a matrix $A$ over a PID in an intrinsic way by relating them to the GCD of the $k \times k$ minors of $A$ for all $k$ . Actually, since the GCD isn’t defined for general rings, we will instead consider the ideal generated by the $k \times k$ minors (which makes sense for any ring, and is the ideal generated by the GCD in the case of a PID).

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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:

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 $M \cong R^r \oplus R/(d_1) \oplus R/(d_2) \oplus \cdots \oplus R/(d_n)$ .