2025-11-10

This following is a very brief summary of what happened in class on 2025-11-10.

We began by finishing the proof of the structure theorem for free modules over a PID. That took most of the class period, but we did have time to note the immediate corollary, namely the structure theorem for finitely-generated modules over a PID.

After first expressing the general module $M$ as a quotient of a free module and then using the structure theorem for free modules, we concluded that there is an integer $n \geq 0$ , nonzero nonunit elements $a_{1}, \dots, a_{m} \in R$ satisfying $a_{1} ∣ a_{2} ∣ \dots ∣ a_{m}$ , and an $R$ -module isomorphism

M ≃ R / ⟨ a_{1} ⟩ \oplus \dots \oplus R / ⟨ a_{m} ⟩ \oplus R^{n}

The number $n$ was then defined to be the free rank of $M$ , while the elements $a_{1}, \dots, a_{m} \in R$ were defined to be the invariant factors of $M$ . (They are unique up to multiplication by units).

Next class we will specialize to the case $R = F [x]$ with $F$ a field. In doing so, we'll obtain some famous fundamental results of linear algebra!

Concepts

References

Dummit & Foote, Abstract Algebra: Section 12.1