TITLE: Module Theory for the Working Student
QUESTION [11 upvotes]: Question:

What level of familiarity and comfort with modules should someone looking to work through Hatcher's Algebraic Topology possess?

Motivation:
I am taking my first graduate course in Algebraic Topology this coming October. The general outline of the course is as follows:

Homotopy, homotopy invariance, mapping cones, mapping cylinders
Fibrations, cofibrations, homotopy groups, long-exact sequences
Classifying spaces of groups
Freudenthal, Hurewicz, and Whitehead theorems
Eilenberg-MacLane spaces and Postnikov towers.

I have spent a good deal of this summer trying to fortify and expand the foundations of my mathematical knowledge. In particular, I have been reviewing basic point-set and algebraic topology and a bit of abstract algebra. My knowledge of module theory is a bit lacking, though. I've only covered the basics of the following topics: submodules, algebras, torsion modules, quotient modules, module homomorphisms, finitely generated modules, direct sums, free modules, and a little bit about $\text{Hom}$ and exact sequences, so I have a working familirity with these ideas. As I do not have much time left before the beginning of the semester, I am trying to make my studying as economical as possible. So perhaps a more targeted question is:

What results and topics in module theory should every student in Algebraic Topology know?

REPLY [7 votes]: I read Algebraic Topology by Hatcher before I really knew what it meant for a module to be flat, or projective . I of course knew what a module was, but besides that, I don't think there's much more needed.
The arguments (as I remember them) in the books where you need to know about modules comes from homology or cohomology. It is crucial that you know what the kernel of a map is, and can spot short exact sequences (and know why they're useful). 
Maybe there's some deeper theorems in the chapter on cohomology, but I think you could probably try to learn that as you go along. 
So, in short, I think you should just go ahead and read it. If you encounter problems, just ask here and we'll be sure to help you!