Скачать или смотреть Generating Sets for Free Modules and F[x]-Modules (Algebra 2: Lecture 17 Video 3)

Lecture 17: We started this lecture by defining the submodule of an R-module M generated by a subset A of M. We defined what it means for a module to be generated by a subset. This led to the definition of what it means for an R-module to be finitely generated and what it means for an R-module to be cyclic. We discussed R-linear combinations of a set of elements of a module. We gave several examples including Z-modules, a ring R as a (left) module over itself, and R[x] as both an R-module and as an R[x]-module. We talked about generating sets for the free module R^n and the related example R^{infty}. We discussed when an F[x]-module V is cyclic.

Reading: Our main reference for this lecture was the beginning of Section 10.3 of Dummit and Foote (pages 351-353). We also gave some definitions and examples from Conrad's Introductory Notes on Modules (pages 2-3): https://kconrad.math.uconn.edu/blurbs....