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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 :

  1. Rings, ideals, quotient rings, homomorphisms and isomorphisms, prime ideals, maximal ideals, integral domains, field of fractions. (recap)
  2. Modules, submodules, quotient modules, morphisms, tensor product, flat modules.
  3. Chain condition.
  4. Completion, localization, valuation.
  5. Regularity, introduction to dimension theory.
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    $\begingroup$ If there is a syllabus, and you are taking the course, then why exactly do you need book recommendations? Why can't you just ask the professor? Or just find whatever book interests you, if it will only be a supplement for the course? $\endgroup$ Jan 3, 2021 at 10:45
  • $\begingroup$ @MorganRodgers this course will be done online and no regular classes due to the pandemic, so I have to rely mostly on self-study. And also I want to start early. I just wanted to know if anyone here have studied these topics and can help me in selecting a good book. $\endgroup$ Jan 3, 2021 at 11:01
  • $\begingroup$ Most of these topics are in Dummit & Foote. Lang would fill in any gaps. The two books by Knapp would be an alternate that might be more approachable than Lang, depending on your level. But it will be pretty specialized to find a single book that covered all of these exact topics. I would reach out to your professor, it can be good to get in contact and let them know you are interested. $\endgroup$ Jan 3, 2021 at 11:10
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    $\begingroup$ What have you studied in Algebra 2 (and 1)? Algebra 3 looks like a basic introduction to commutative algebra, so you should look for books on commutative algebra. Big names are Atiyah-Macdonald, Eisenbud, Matsumura and e.g. the notes by Gathmann on Commutative Algebra are also excellent (as are all of his notes). All of these topics are covered in commutative algebra books. $\endgroup$
    – Qi Zhu
    Jan 3, 2021 at 11:11
  • $\begingroup$ The terms "Algebra 1, 2, and 3" are localized course titles, and not universally understood terms with any mathematical meaning. I have edited your question to make the language a little more universal. $\endgroup$
    – Xander Henderson
    Jan 3, 2021 at 23:04

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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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