2025-11-06

This following is a very brief summary of what happened in class on 2025-11-06.

Our new goal is to find a structure theorem for modules. This structure theorem will boil down to expressing a given $R$-module $M$ as a direct sum of some "simple" $R$-modules. This decomposition will be basically constructive, but that construction will require $M$ to be finitely generated in order for the process to complete. Our construction will also be inductive, and hence require submodules of $M$ to also be finitely generated. You might think that all submodules of a finitely-generated submodule are automatically also finitely generated, but we looked at an example to show that this is not necessarily the case.

To address this, we introduced a finiteness condition on modules, namely the property of being Noetherian. In short, this is the ascending chain condition on submodules, but it is also equivalent to the property that every submodule be finitely generated. As a special case, when a ring is viewed as a left module over itself (and so submodules correspond to ideals), this condition becomes the ascending chain condition on ideals (and is equivalent to the condition that all ideals are finitely generated).

We then gave slightly more detail to our upcoming strategy, namely that we'll use the finite generation to start with a surjection $\pi :F(X)\to M$ from a free module on a finite set $X$. By the First Isomorphism Theorem, this will tell us $M\simeq F(X)/\ker (\pi )$. This reduces our problem to understanding quotients of free modules.

We then turned our attention to free modules $M$ with a submodule $N\subseteq M$. Our goal is to find "bases" for $N$ and $M$ that are "aligned", so that we have a nice description of the quotient module $M/N$.

Finally, we made that statement precise in the structure theorem for free modules over a PID. This theorem will be the cornerstone of the upcoming Fundmental Theorem for Modules over a PID. We will spend all of next class walking through the proof of the structure theorem for free modules.

Concepts

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

References

• Dummit & Foote, Abstract Algebra: Section 12.1