Can I skip part III in Dummit and Foote? 
I am reading Abstract Algebra by Dummit anfd Foote. I have already taken an introductory course in linear algebra, mostly at the level of Strang's MIT OCW Linear Algebra. My question is: Can I skip 'Part III: Modules and Vector spaces' and jump directly to chapter 13: Field Theory in Dummit and Foote? Will that be harmful? 

(Note that I am planning to cover Part I up to 5.3 and Part II up to 9.5, as suggested in preface of D&F. )
 A: It depends what your goal is. If you want to get to the culmination of a classical abstract algebra class, Galois theory and the solvability of the quintic, you will not be missing much if any.
If you plan to go on to graduate study (even in group theory or field theory!), or even read through the rest of D&F, then you will in all likeliness encounter material that build on PID modules.
However there is no harm in skipping part III for the time being. A lot of the abstract module business is more abstract than you have encountered so far and will benefit from having a motivation to read it.
Also make sure that you do not just read the chapter text but also look at the exercises. The standard result and applications of the theory -- matrix normal forms and finding particular conjugating matrices -- are hidden in the exercises of these chapters.
(Basis for my remarks: I have taught multiple times a first-year graduate class on this material, using D&F)
A: You should be fine doing this. The proofs in field theory mostly use more basic notions from linear algebra. So if you feel comfortable with vector spaces and linear transformations, then this is ok. (In fact, the graduate level abstract algebra sequence I am taking is doing this)
