Do commutative algebra only if your goal is algebraic geometry. Two popular books I don’t know are one by Aluffi and another by Michael Artin. Also, consider learning another subject. If you’ve already learned multivariable calculus, consider Differential Topology by Guillemin and Pollack. Or a book about the differential geometry of curves and surfaces.

Do exercises from those books.

If you haven't done much in discrete mathematics or linear algebra, you might want to study those to get a feel for areas where abstract algebra is applied a lot. There are plenty of good texts available, and some of the good ones are free. Also, linear algebra has applications to other areas of algebra. Representation theory, in particular.

For discrete mathematics I recommend A Cool Brisk Walk Through Discrete Mathematics. It's a nice text, and the author has made it available free online. Hard copies are sold, too. I also find it pleasant for leisure reading, having passed my undergraduate courses in discrete math decades ago.

I liked Visual Group Theory by Nathan Carter as a supplement to help solidify my feeling for group theory.

Since you expressed some interest in commutative algebra, here's a couple you might want to put on the back burner for motivation: Cox, Little & O'Shea - Ideals, Varieties, and Algorithms is a nice introduction to algebraic geometry. Eisenbud's Commutative Algebra is intro graduate level, but if you can, say, borrow a copy through interlibrary loan, I recommend reading some of the chapter introductions and other prose to give you a "peek over the mountain," as it were.

What level do you recommend for read theory in order to pas rla ?

Herstein's Topics in Algebra is a good introduction and tends to explain concepts using words rather than equations. He makes difficult topics easy to follow, but the book is fairly dated and doesn't use modern notation, and doesn't go into nearly as much depth as you'll see in Dummit and Foote.

I know that this sub likes to recommend Pinter, and I like to promote freely available resources, like Abstract Algebra: Theory and Applications by Judson.

I like Dummit and Foote, but that’s just me, and probably isn’t the best first time intro to abstract algebra

For an introductory text, I recommend Herstein's Topics in Algebra. It slowly walks you through groups, rings, vector spaces, modules, fields, linear transformations and other selected topics.

Has plenty of exercises and doesn't skip over any details.

Gallian's book is a pretty good place to start.

He's pretty hand-holdy and does a TON of examples

I remember Pinter being widely recommended.

I got mine in PDF, Abstract Algebra 7Ed (Fraleigh) it’s easy to understand and there are a lot of exercises to work on. :)

Life pro-tip: every book becomes highly affordable when you use library genesis. But of course I’d never recommend something like that.

Abstract Algebra by Durbin gets recommended by my university. Not brilliant but it likely covers what you’re looking for

As this is basic abstract algebra, there is no need to get a book for that. Lecture notes would do equally well. For instance, look at

https://math.berkeley.edu/~apaulin/AbstractAlgebra.pdf

If you want to go further than that, just let me know.

Judson's text is available free online: http://abstract.ups.edu/download.html

Hungerford.

Kidding, of course. What I really meant was Hungerford. He has two books. Try the introductory one.

Another good one is Gallian.

Late undergrad to beginning grad you can try Papa Hungerford, Dummit and Foote, Aluffi, and Lang.

I study better with a physical book so I recommend A Book of Abstract Algebra by Pinter

The level of undergraduate algebra courses varies significantly. Given the time passed, I don't think a graduate level text would be suitable.

A Book of Abstract Algebra by Pinter is very accessible and it's available as a Dover reprint. A First Course in Abstract Algebra by Fraleigh is a popular prescribed text for undergraduate students, as is Contemporary Abstract Algebra by Gallian.

There's also Abstract Algebra: Theory and Applications by Judson, which is available free online.

Abstract Algebra by Dummit and Foote is the standard prescribed text for graduate students. It's very comprehensive and it has lots of examples and exercises.

Personally, I like Basic Algebra by Knapp, which is available for free here. Alternatively, you could try Basic Algebra I by Jacobson, which is available as a Dover reprint.

Lang's Algebra (GTM) is famously difficult to learn from (though an excellent reference), but his Undergraduate Algebra (UTM) is very accessible.

Aluffi's Algebra: Chapter 0 is excellent, but I don't think it's suitable. I haven't read it yet, but his other book, Algebra: Notes from the Underground, is also well reviewed at the undergraduate level.

I'd also mention that with a lot of graduate level algebra topics, you could be better served by buying texts specific to the individual topics.

CC'ing my answer to a related question.

• Algebra: Contemporary Abstract Algebra (Gallian) for an example-rich introduction or Carter has a nice Visual Group Theory book too. though Algebra and Geometry (Beardon) shows how interconnected the areas of maths are (see my block quote here for a teaser). Since you studied it before, you might find a (more encyclopaediac, but also more terse) text like Lang useful too.
• Proofs and Fundamentals (Bloch) if you need a recap on logic and proof techniques.

I got my introduction to abstract algebra my senior year in college, 1985 it was, and by mysterious vicissitudes of fortune, my textbook has stayed with me all this time. It was Fraleigh's third edition. I recently picked it up again and started working through it. My rusty math gears don't work quite like they used to. I've hunted around for what would be a superior text for self study, and nothing has stood out. What I'd really like is a way to check my work on the exercises.

Since you have a suitable background in algebra you can try

S. Lang Algebra

Jacobson, Basic Algebra

Grillet, Abstract Algebra

I’m going to recommend a book called Groups, Rings, and Fields, by DAR Wallace. It is part of the Springer undergraduate mathematical series (SUMS). You should be able to get a used copy for just a few dollars. One warning: there are some typos in it, but they are very easy to spot, and they don’t present any problems.

Despite doing most of my work in analysis, abstract algebra really was my favorite.

Have fun, best wish 🙋🏻‍♂️

I ran into this site; https://yutsumura.com while studying for quals. It's undergrad level more than grad level.

However, a lot of the time there are no "steps" in an abstract algebra problem, and the solution is the steps. A proof provides all the important details why it is true. If you have a hard time understanding some solutions, feel free to post them here; people can help work through the solutions with you. Based on what you've said, a good goal in this class for you, should be to reach the point where you can read the solutions and see why they are true, usually without needing someone else to give you clarification. You might need to go back to your text and reread the definition or the statement of some theorems, though, and that is fine.

Knapp has two books "Basic Algebra" and a follow-up "Advanced Algebra" where the goal is to cover all the algebra "every mathematician should know".

So they are very complete, covering essentials from elementary number theory, linear algebra, multilinear algebra, groups, rings, galois theory, modules and then in the advanced book going more into modern number theory, homological algebra and algebraic geometry.

Super versatile books, probably the most complete textbook in Algebra, though not sure about the geometric context or the examples as you prefer them.

It sounds like Algebra by Michael Artin would be right up your alley. He is an algebraic geometer and brings a long of linear algebraic and geometric context. And it also covers a wide swath of modern abstract algebra.

I think if u already know some basic algebra u should probably start reading Hartshorne algebraic geometry, it's essentially a geometry book with lots of examples and exercises, it goes beyond undergrad curriculum though it's quite easy and readable as an undergrad, and it doesn't require too deep prerequisite knowledge, while involving lots of algebra and building algebraic geometry from ground up. But if u encounter any difficulty in the book, say if there is something in algebra that u don't understand then u can refer to some standard text to learn that, or ask your prof. But there are some important things (though elementary) that aren't mentioned in the book and that could be a problem, but it's a great book in general and will allow u to learn algebra and algebraic geometry and topology with relative ease, and if u want to study algebraic geometry u will have to read it eventually anyway. So it is easy, covers lots of subjects including algebra but none too deeply and is quite versatile. It has lots of geometric context also obviously.

Edit: Why is everybody downvoting me? OP asked for a geometry based book that has a lot of algebra as well (OK maybe Hartshorne doesn't have algebra directly but it requires a lot of it and also involves some category theory indispensable to any algebraist or algebraic geometer) and doesn't dig too deeply on any topic. So I thought reading it would be a good idea. Also I don't understand why Hartshorne gets so much hate, it's like if u know some basic stuff and want to skim through geometry it is still the best resource, Mumford's red book is too detailed and functorial for a beginner, u need to be quite categorified already, and what else, read EGA??? That would be crazy! So Hartshorne even today is the best resource for beginners.

Lang

Probably Dummit & Foote would be a great read for you. After the basics of two-semesters of undergrad algebra, it also discusses some homological algebra, module theory, has a quick intro to commutative algebra etc. I like their writing style too. P.S.: you definitely need to be doing a lot of their exercises... or else you're actually missing out on some important related concepts.

You have to be patient with this book but imo, every was worth a read.

I agree that doing exercises without feedback isn't as helpful as you'd want it to be. I'd be more than happy to look over anything you'd feel like sharing. I know we've started a discord from r/math just for study of a particular book, but I'd be more than happy to set one up or participate in one for general learning! Love that you're pushing yourself to learn more. Let me know what works for you.

How about studying Rubik's cube instead? You can learn most of the basic group theory through that while building a program that impllement the ideas. Not to mention picking up a party trick. There several books analyzing the cube on different levels.

I have this book by David Joyner called Adventures in Group Theory subtitled, Rubik's cube, Merlin's Machine, and Other Mathematical Toys. It is designed sort of for people generally interested in mathematics. So non-professional math. The tone is light.

It doesn't develop a solver but any programming language with a full set of operators on lists and sets can be used to create the operators necessary to explore the Rubik's group structure fully as discussed, instead of just reading the theorems and the proofs. I find learning generally easier if you have a goal in sight.

Another really intereting group theory puzzle to develop on is Polya Pattern Ananlysis. You can find examples and codes for it in the book Mastering Mathematica by John Gray. It is a great read to test your understanding of group theory.

Try A Book of Abstract Algebra by Charles Pinter. It’s not too rough. It’s exactly what my advisor recommended after I asked him the same question.

Dummit and Foote is the best text to introduce abstract algebra in my opinion.

Artin's Algebra is what we use for Algebra, which I like a lot. 6.042 (Discrete Math) is really a class for learning to prove things - if you struggle with proofs, How to Prove It and Book of Proof are good starter resources I used to skip 6.042.

A student here wrote an awesome book with around 100 pages of abstract algebra that I'm using to self-study along with Artin's book. You can find it here. It has quality content like this.