Classification of Modules over a PID: Elementary Divisors (Algebra 2: Lecture 26 Video 3)

Lecture 26: We started this lecture by recalling the Classification of Finitely Generated Abelian Groups. We then recalled some material about F[x]-modules. In particular, we saw that a finitely generated Z-module is isomorphic to a direct sum of finitely many cyclic Z-modules, so we spent some time talking about cyclic R-modules. We saw that for R a PID, a cyclic R-module has a particularly nice form. We stated the main theorems for this classification result; the existence part in both invariant factor and elementary divisor form, the primary decomposition theorem, and the uniqueness statements. We saw that specializing to R = Z recovers the Classification of Finitely Generated Abelian Groups.

Reading: We first reviewed some material from Section 5.2. I recommend looking over this if you have not thought about it for a while (this will help you to understand the proofs in this section). We also reviewed some material about cyclic R-modules from Section 10.3. We closely followed Section 12.1 for the statements that we gave in this lecture. In the next lecture we will go back and give proofs of some of these results following the presentation in Section 12.1.

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