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I've been teaching myself algebra these couple of months. I already went through the basics of group (Lagrange, action, class equation, Cauchy and Sylow theorems etc.) And I already have some linear algebra background.

I started to read about rings and got really excited.

Is there any book you can recommend to me for learning ring theory from the basics with a view towards algebraic geometry/topology?

(Locally ringed spaces seem extremely interesting and I'd like to approach them as soon as possible.)

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I would recommend Atiyah-MacDonnald. –  JPLF Jan 5 at 1:01
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If you've only just learned some group theory I cannot recommend Atiyah-MacDonald. If you'd like to learn commutative algebra (and it sounds like you would!) you'll need to learn the basics first. Popular texts are Herstein and Artin for an introductory book, and Hungerford and Lang (I love Hungerford) once you've gone through one of the former. –  Mike Miller Jan 5 at 1:02
    
I find myself really fond of Gertrude Ehrlich's "Fundamental Concepts of Abstract Algebra". –  snulty May 1 at 20:21

7 Answers 7

up vote 9 down vote accepted

Another remarkable book is Miles Reid's Undergraduate Commutative Algebra.
It is quite elementary (as the title indicates) and very short: 153 pages.
It is written by a renowned algebraic geometer for budding algebraic geometers.
It is chock full of pictures showing how to interpret geometrically algebraic notions: just look at the frontispiece of the book, which you can see in the link I gave above.
That frontispiece is a very realistic picture of a module $M$ lying over its base ring $R$, illustrating in an amazingly visual way the maximal points of the support of $M$, the stalks of $M$, the generic point of $R$, etc.
Already in Chapter 0 (called "Hello!": the author has a very friendly and amusing style) you will find pictures of the cuspidal cubic and of $\operatorname {Spec} \mathbb Z[\sqrt -3]$, hinting at the amazing synthesis between geometry and arithmetic permitted by scheme theory.

In a nutshell, that very elementary book exactly addresses the OP's wish to learn ring theory "with a view towards algebraic geometry" .

Edit
Since I have also recommended Atiyah-Macdonald's book, how do both books compare?
Here is Miles Reid's point of view (page 12):
"[Reid's] book covers roughly the same material as Atiyah and Macdonald, Chaps. 1-8 but is cheaper, has more pictures, and is considerably more opiniated.

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For general algebra my favorite is Abstract Algebra by Dummit and Foote. It's very very accesible for beginners but covers a huge amount of material.

After getting comfortable with general abstract algebra and you want to move to commutative algebra with a goal of learning algebraic geometry (which I would assume from your tags) then I'd recommend Commutative Algebra: with a View Toward Algebraic Geometry.

Both of these are books you really need to go through some of the exercises when reading but if you do that you will get a lot out of them.

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thanks! should i finish all of D&F before approaching commutative algebra or homological algebra? –  user116395 Jan 5 at 16:33
    
I'd say go through most of Part II,III, and IV (ring theory, module theory, and field theory) and the first chapter of part V (some background to commutative algebra). –  Dori Bejleri Jan 5 at 17:23
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Dummit and Foote is a wonderful book. –  Ryan Reich Jan 5 at 20:18

Kaplansky's Ring Theory book is great.

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Atiyah-Macdonald has been the best introduction to commutative algebra from the moment it was published in 1969.
Actually I think it is one of the most extraordinary textbook ever published in all of mathematics .
It is exactly 128 pages long, hence also one of the thinnest mathematics books on the market, but contains a mind-boggling quantity of material.
It starts with the definition of a ring (!) on page 1 but already in the exercises to Chapter 1 you will find a self-contained introduction to affine algebraic geometry, both classical and scheme-theoretic (and as an aside, remember that schemes were very new in 1969).
The book calmly goes on to chapter 11, the last one, where different definitions of dimension are given but proved to be equivalent.
You will also learn in that chapter about Hilbert functions and regular local rings, two notions which play a great role in algebraic geometry.
I won't even try to summarize the other chapters: suffice it to say that every basic notion in commutative algebra is covered: the Nullstellensatz for example is proved (or given as an exercise with hints) several times.
And the most remarkable feature of the book is that every proposition is proved, crisply but completely, without cheating or resorting to hypocritical shortcuts like "it is easy to see..." or "it is left as an exercise ..."

There are other good books on commutative algebra: Bourbaki, EGA, Eisenbud, Patil-Storch, Zariski-Samuel, ... but they are probably too advanced for a beginner, whom they might discourage rather than help.
I advise you to to use them as reference books once you have studied a reasonable part of Atiyah-Macdonald.
Good luck!

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I agree with you, of course, but probably Atiyah-Macdonald is not the best first introduction to ring theory. I think the authors assume some familarity with the notion of rings, or at least assume some mathematical maturity. The reason is that the proofs are quite short (though, complete) and many stuff is covered by the exercises, which other books would include into the text. That being said, I think you can become a good mathematician when you (try to) solve all the exercises in this book. –  Martin Brandenburg Jan 5 at 10:07
    
@Martin. Yes, you are right: although technically the authors assume only few previous results, the book definitely presupposes a certain maturity of its readers. So indeed they would read the book with more profit if they were already comfortable with the main notions in a first course in general abstract algebra, as developed in Mike Artin's Algebra for example (although reading that book in its entirety is certainly not required). –  Georges Elencwajg Jan 5 at 10:48
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Artin's "Algebra" is a great suggestion. –  Adeel Jan 5 at 14:31
    
I agree with Martin: I would never recommend Atiyah-MacDonald as an introduction: it is far too terse and its choice of proofs to give reflects a kind of high-level thinking that would confuse the beginner (see, for example, what they pass off as Nakayama's lemma). Working your own way through it without any ring theory background would be valuable to someone highly talented, and inefficient for almost anyone else. –  Ryan Reich Jan 5 at 20:20

I recommend P.M. Cohn's book: Introduction to Ring Theory. It has the advantage of being clear and concise (just over 200 pages).

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Dan Saracino's Abstract Algebra is a good way to get acquainted with the early concepts of ring theory.

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