2024-11-15

This following is a very brief summary of what happened in class on 2024-11-15.

We began by recapping the structure theorem for free modules over a PID. We then proceeded to consider the more general case of a finitely-generated (but possibly not free) module over a PID. After first expressing such a module as a quotient of a free module and then using the structure theorem for free modules, we were ultimately able to deduce that there is an integer $n \geq 0$ and nonzero nonunit elements $a_{1}, \dots, a_{m} \in R$ satisfying $a_{1} ∣ a_{2} ∣ \dots ∣ a_{m}$ such that there is an isomorphism

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

We mentioned (but didn't prove) that this decomposition is unique "up to units." 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).

We then briefly explained the special case in which $R = Z$ , in which we recover the Fundamental Theorem for Finitely-Generated Abelian Groups.

We also discussed the idea of factoring each $a_{i}$ into a product of prime powers, obtaining the so-called elementary divisor decomposition of $M$ .

Next week we specialize to the case $R = F [x]$ with $F$ a field. We'll obtain some famous fundamental results of linear algebra!

Concepts

References

Dummit & Foote: Section 12.1