TITLE: Book recommendation for a third course in abstract algebra
QUESTION [1 upvotes]: I completed an abstract algebra course on groups, and a second course on rings and fields a few semesters back and I want to take a third course next semester. For the previous two courses, I followed mainly I. N. Herstein and J. A. Gallian books (and also looked into Dummit & Foote and J. Rotman, but not in detail).
For the third course, I'm a little confused and I need some book recommendations. The syllabus covers these topics :

Rings, ideals, quotient rings, homomorphisms and isomorphisms, prime ideals, maximal ideals, integral domains, field of fractions. (recap)
Modules, submodules, quotient modules, morphisms, tensor product, flat modules.
Chain condition.
Completion, localization, valuation.
Regularity, introduction to dimension theory.

REPLY [0 votes]: Dummit and Foote will serve you in good stead for a long time in algebra, although no book is certainly perfect. With the exception of their treatment of tensor products which I am not a fan of, I think the rest of their explanations are quite nice and good for self studying. Ultimately, assuming cost is not an object (or if you are getting these from a library or libgen) it is helpful to have a few different books available so you can look to a different exposition of a topic if you get stuck. Having multiple books will also help expose you to a greater variety of notation, which while confusing at first will be helpful in the long run.
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