Introduction to Modules (Algebra 2: Lecture 13 Video 1)

Lecture 13: In this lecture we began our discussion of modules.  We started by recalling what it means for a set V to be vector space over a field F.  We then defined what it means for a set M to be a left R-module and we saw that modules over a field F and vector spaces over F are the same thing.  We defined what it means to be a submodule.  We saw that a ring R is a left module over itself and the submodules are the left ideals of R.  We defined the affine n-space over a field and the free R-module R^n.  We discussed Z-modules and saw there is exactly one Z-module structure on any abelian group.  We defined what it means for an ideal to annihilate a module and saw that if I annihilates an R-module M, we can give M the structure of an (R/I)-module.  We discussed the example of elementary abelian p-groups.  We briefly discussed some additional examples, for example, R[x] is an R-module, and Map(R,R), the set of all functions from R to R, is an R-module.