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I've been teaching myself basic (abstract) algebra. I'm not studying for any particular subject. I just want to learn anything that might be useful in the future. Since I have enough time, I treat each part separately. For example, for group theory I'm reading Rotman's An Introduction to the Theory of Groups besides books on abstract algebra. Now I think I know enough group theory to start reading ring theory. I also know basic linear algebra.

I'm looking for an introductory ring theory text on a level similar to that of Rotman's group theory book. However, I don't know what topics a ring theory text shoud include, though I do think that commutative algebra might be too advanced for me now. I have looked through many questions here, e.g. 1, 2, 3, and there are a lot of books listed that look good. However, I'm not sure which book I should read.

I would like to find a book with the following:

  1. It is introductory and starts from the basics. It does not assume that the reader is familiar things like category theory or homological algebra. It's best if the book spends some time on introducing these.
  2. It is thorough and comprehensive. I'm a fan of thick books, so I actually like lengthy (but not wordy) books that contain lots of material.
  3. It is a book on general ring theory (not only commutative algebra).

And please don't recommend texts on abstract algebra, since I already know some good books. I plan to read abstract algebra books concurrently with ring theory texts. :)

Thanks in advance!

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  • $\begingroup$ It is really not possible to have a comprehensive book in a branch of mathematics anymore. But I guess you mean "comprehensive about the basics" maybe. An introduction is usually necessarily not comprehensive, so I guess this is the only interpretation that makes sense for you. $\endgroup$ – rschwieb Nov 19 '17 at 12:09
  • $\begingroup$ Anyhow, many texts on abstract algebra contain a "comprehensive introduction" to ring theory. For example, Martin Isaacs' Abstract algebra, and Jacobson's Basic Algebra I+II. So you are simply undercuting yourself by declining such suggestions. $\endgroup$ – rschwieb Nov 19 '17 at 12:10
  • $\begingroup$ @rschwieb Well, I'd like to use the adjective "comprehensive" when a book contains a lot of material... The kind of book I have in mind is similar to that of Rotman: It contains (much) more material than abstract algebra books and yet starts from the basics. I plan to read some books on abstract algebra concurrently with some books for ring theory only, as I've been doing for group theory. $\endgroup$ – Colescu Nov 19 '17 at 12:32
  • $\begingroup$ @rschwieb Then I think I should read abstract algebra texts first. Thank you for your suggestion! $\endgroup$ – Colescu Nov 19 '17 at 12:41
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Since you're looking for a book of an "introductory" level and which starts from the basics I think you should have a look at the book "A first course in Rings and Ideals" by David Burton.

This book covers the basics of ring theory, e.g., maximal and prime ideals, isomorphism theorems, divisibility theory in integral domains, etc; and also includes some topics of commutative algebra (chapter 12: "Further results on noetherian rings"), as well as, noncommutative algebra (chapter 13: "Some noncommutative theory").

On the other hand, because not only the theory is important, but also the practice in order to master a subject, I would suggest you to check the book "Exercises in Basic Ring Theory" by Grigore Calugareanu and Peter Hamburg. This book, as the title says, its aimed to help you to understand the basics of ring theory by offering a nice set of exercises from the fundamentals to rings of continuous functions. I recommend you to use this book as a companion of Burton's book.

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T. Y. Lam's A First Course in Noncommutative Rings: It pretty much starts from the basics without using category theory, has lots of examples and a nice set of exercises. The author has also written a second volume, "Lectures on modules and rings" that covers more on modules and the homological aspects.

Anderson and Fuller's Rings and Categories of Modules is category theoretic introduction to noncommutative ring theory but it might not be very useful as a first introduction to the subject.

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  • $\begingroup$ Thanks! I've heard of Lam's book, and it looks pretty good. What prerequisites do I need for this book? $\endgroup$ – Colescu Nov 19 '17 at 9:12
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    $\begingroup$ @Colescu Probably just the level of mathematical understanding it takes to read and write proofs. It is a great book for noncommutative ring theory. But this book does not include the basics of commutative ring theory, which presumably one would want for a "thorough introduction" $\endgroup$ – rschwieb Nov 19 '17 at 12:13
  • $\begingroup$ @rschwieb I tried to read the book but I couldn't get through the first section... The material is a bit too advanced for me, although I have already read about the basic notions (the chapter on ring theory in Hungerford's Algebra text). I think I should read it later. Anyway I think it's really a somewhat exaggeration to say that it has such minimal prerequisites... $\endgroup$ – Colescu Dec 3 '17 at 14:48
  • $\begingroup$ @Colescu yes, I would recommend it as a great second book (after a basic introduction) $\endgroup$ – rschwieb Dec 3 '17 at 15:36
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Paolo Aluffi's Algebra Chapter 0 covers ring theory pretty well, but it's not just about ring theory. It also does not assume that reader is familiar with category theory, but it will be one of the first things in the book. Second edition was just released.

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