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.