F[x]-Submodules (Algebra 2: Lecture 14 Video 3)

Lecture 14: In this lecture we discussed F[x]-modules. We first saw why an F[x]-module V had to be a vector space over F. We then saw that the action of the polynomial x on V determines the entire F[x]-module structure. We then saw that the action of x on V had to be a linear transformation. That is, we saw that a vector space V over the field F together with a linear transformation T: V to V gave V an F[x]-module structure. We saw some examples including one from the 'shift operator' on affine n-space over F. We then discussed F[x]-submodules. At the end of the lecture we defined what it means for a map to be an R-module homomorphism. We can now ask, 'When are two F[x]-modules isomorphic?' This question motivates some of the linear algebra that will be a focus of the end of the beginning of Math 206C.

Reading: We pretty closely followed the presentation in Section 10.1 of Dummit and Foote, pages 340-342. At the end of the lecture we referred to some things that are coming up later in the course in Sections 11.2 and 12.2.