2024-11-14

This following is a very brief summary of what happened in class on 2024-11-14.

We briefly recapped the notions of Noetherian and rank. We noted that a ring $R$ (when viewed as an $R$-module) is Noetherian exactly when all of its ideals are finitely generated; in particular, every PID is Noetherian. We also noted that for a free module $M\simeq F(\{x_1,\dots ,x_k\})$, the corresponding generating set $\{m_1,\dots ,m_k\}$ is linearly independent (and hence is worthy of the name "basis").

We then spent the rest of the class working through the main steps in the proof of the structure theorem for modules over a PID. Be sure to glance at the notes for a full proof that covers every detail.

Next class we'll look at the fundamental theorem for modules over a PID.

Concepts

• Noetherian modules
• Linear independence, rank and the structure of free modules

References

• Dummit & Foote: Section 12.1