Fitting ideals of modules

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.

Hans Fitting

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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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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