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 over a PID in an intrinsic way by relating them to the GCD of the minors of for all . Actually, since the GCD isn’t defined for general rings, we will instead consider the ideal generated by the minors (which makes sense for any ring, and is the ideal generated by the GCD in the case of a PID).