2024-11-12

This following is a very brief summary of what happened in class on 2024-11-12.

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 proceeded to introduce the notion of linear dependence in modules, defined identically to how it is defined in vector spaces. When $M$ is a free $R$-module, a set of generators for $M$ is always linearly independent. We then proved that, at least when $R$ is an integral domain, in a free module on $n$ generators that maximum size of any linearly independent set is $n$. This led us to more generally define the rank of an $R$-module as the maximum size of any linearly independent set in the module. For free modules, this number matches the number of generators in a "basis" for the free module; in the case $R=F$ is a field, this number matches the dimension of the $F$-module (i.e.., $F$-vector space).

Finally, we stated 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: Section 12.1