Noetherian R-Modules (Algebra 2: Lecture 27 Video 1)

Lecture 27: We started this lecture by defining what it means for an R-module to be Noetherian.  We gave several equivalent characterizations of Noetherian R-modules.  We then defined the rank of an R-module.  We proved that in a free module of rank n, and n+1 elements are linearly dependent.  We then stated a theorem about submodules of free R-modules when R is a PID and showed how to use it to prove the Existence statement in Invariant Factor Form for the Classification of Modules over a PID.  At the end of the lecture we started the proof of this remaining theorem.

Reading: In this lecture we very closely followed Section 12.1 of Dummit and Foote.  We proved Theorem 1, Proposition 3, and Theorem 5 (assuming the statement of Theorem 4).  We started the proof of Theorem 4 and got up to the end of the first paragraph on page 461.

[speedbird]

At first it seems really strange that an R- module is itself finitely generated but its R-submodules not be finitely generated.