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.
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.