Artin's book is probably one of the better books, more because of the way you have to read it to learn it. Artin's book is extremely nonstandard, in the sense that it isn't so "encyclopedic" as you usually encounter with the whole theorem, corollary, proof, proof, proof, example, example sequence. What I think a lot of readers miss is that Artin's book makes you fill in the details he leaves out by using the hints he mentions in words within the text. For example, I was able to expand the two pages of notes on Ch 2, section 5, in Artin into about 8 pages of original notes and theorems, just by digging for the main points. If you want a sample of my notes, please email me and I'll email you a brief PDF sample for you to compare. That being said, assume that you will have to dig a lot in this book, and should you choose to study from it, I suggest the following:

How to read it:

With a cup of coffee, or tea, and a notepad of paper for you to make comments on. Do not take notes; anyone knows that simply rewriting things doesn't do anything for learning. You should do the proofs in different ways, if you can see how, and try to make some of the aside remarks he makes into theorems or more precise ideas (this is not to say that Artin lacks rigor; this is just talking about the general commentary. When he makes commentary, it always seems to be enough to actually dig out exactly what to do after a little scratching). He also leaves a lot of easier proofs to the reader, so do them.

Is non-standard a less-rigorous approach?

No. Artin is definitely doing his own thing here, but I think it works really well. Getting through that book FORCES you to take responsibility for your math education by making you get your hands dirty while also developing an intuitive understanding of algebra.

What about his personal flavor of algebra?

Well, it's fairly clear to all of us that texts seem to have different flavors (being a function of the author's research area, and what was fashionable during the time the book was authored). Artin's book is algebra with light strong hints of geometry throughout, as he is in algebraic geometry. You will find that unlike most authors, Artin loves structures made of matrices when working with examples, as opposed to permutation groups or the ``symmetries of the square group,'' known also as the ``octic group.'' While these things have their place in his book, he changes the emphasis here. That's why I suggest using a companion book so as to have two sharply contrasting flavors of presentation, and Herstein seems to write in such a way that would do this. Artin covers a lot of material extremely quickly, but focuses on the bigger picture in several key areas. For example, the sections 7 and 8 in chapter 2 deal almost exclusively with how one would go about investigating a particular group structure to learn about it, teaching a student how to dig into something they might barely understand.

Advice to make a wondeful course:

Use another book which IS encyclopedic as a reference, since Artin doesn't label theorems and definitions so explicitly. I suggest Herstein's Abstract Algebra, or his book Topics in Algebra.

Personal Charracterization:

I place this book as one of my favorites on the bookshelf, and it sits among others like Rudin, Ahlfors, Sarason's notes, Herstein, though it's obvious to me that Artin is on a very very different path than all those books, very nonstandard (Artin DOES DO all that a usual algebra course does, and more, if you were wondering), but as a result, very very very thorough and very clearly presented. I love this book very much.

I agree with the other reviewers in the sense that it is ture Gallian's book is soft on theory and rigor, but oppositely I find this lack of real substance to be Gallian's deepest flaw. I give Gallian one star, basically for effort.

I divide my critique into the following subcategories:

Organization:

Gallian's book is organized well enough in the sense that he opens each chapter with some commentary about the problems to be studied, or motivation, and then proceeds to go example, theorem, proof, example, example, example, example,..., example. This doesn't work, I think, because he spends too little time actually showing theorems and proofs, and sometimes he'll build an entire chapter on just two or three theorems, and fill the rest with useless commentary (which I'll mention again below).

Readability:

As for readability, for people who read math books at all (i.e., those who study outside of class), this book should be a nightmare. If you were to strip away all of the useless commentary/endless biographical insets/weblinks you would be left with probably about 30 pages of theorems and cumbersome proofs (by cumbersome, I don't mean involved, I mean unrefined). Gallian has failed to make a readable text because he presumes to have the omnipotence and foresight required for putting a full understanding of algebra and algebra history into one book. As a result, the excess commentary he makes and useless statements (for example, "In high school, students study polynomials with integrer, rational, real, and sometimes complex coefficients") distract a reader from the main points, and I rarely found myself rubbing my chin thinking how insightful something he said was. All in all, I feel as though the reading felt "hoakie" at best--like he was elbowing me in the side, winking, trying to get me to lie and say I thought what he was saying was insightful.

Exercises:

The exercises are often clumsily put together and the quotes before each problem set can get extremely patronizing. I remember thinking how cocky this Gallian fellow must be to presume that people can't do "his" problems. A joke, to say the least. In any case, they seem fine for all purposes -- if you're going into chemistry or an applied science that uses group theory. It's very obvious that our author believes that group theory is the pinnacle of the algebra experience and struggles to present topics from rings and fields. IF you are someone who likes group theory, fine. BUT Artin's book does everything Gallian does and more with group theory and builds the same ideas on more solid footing, using linear algebra excessively throuhout the book. For example, if you think I'm joking about Gallian's weakness, just look at the chapters on isometries and compare them to the chapters in Artin, and you'll see what I'm talking about.

Peter Rabbit:

Well, I do have at least one nice thing to say. As anyone can see, Gallian has a lot of examples, but this seems to be the only redeeming quality of the book. But that alone doesn't make an algebra book.

Broad Commentary:

If it's a softer touch you're looking for, I'd say go with Durbin -- he's easy to follow and an excellent writer. If it's group theory, examples, and a lot of wonderful exercises you want, go with Artin. Neither of those books get caught up in useless commentary. I've heard good and bad things about Fraleigh (sp?), but have no direct experience with that book. I would suggest, for those who don't want a hardcore book such as Herstein's Topics in Algebra, or M. Artin's Algebra, you should see Durbin's book. Durbin is also a softer book; it has many nice examples and is very well written. IF you are unavoidably made to use this book for a course, and if you want to learn to be more insightful/challange yourself to think/want to study, then I suggest you use any of Artin, Herstein, or Durbin as a companion (in that order, but I only place Artin above Herstein because Artin has more material in it, Herstein is a much better writer so you might choose him depending on which book you'll spend more time with).

then I suggest you use this book for your introduction to analysis. I divide up my critique into the following sections:

Content:
The author of this book expects you to be comfortable with mappings, set theory, linear algebra, etc. I would recommend that you use either Munkres' book on topology, or (if you can't afford that) the Dover book, Introduction to Topology by Bert Mendelson (you should read all of Ch. 3 BEFORE starting Rudin if you want to pick up on which things could be even more general than they are in Rudin - refer to earlier chapters if you don't recognize something). I suggest also looking at continuity in one of the topology books I mentioned. Also, look up the following things and at least know what they are before getting past Ch. 4, so you have some supplemental language to use: Banach space, boundary, basis for a topology, functional.

Like I said, this book is for serious people, and it requires strong focus for you to pick up on all the subtle arguments made through his examples. I do not agree with some people who say this book is bad for an introduction, in fact I think it is the best because Rudin REFUSES to be tied down to single variable concepts which could be explained just as easily in the context of more general spaces. If you are one of those kids who think's you're great at math because you do well in competitions, steer clear; your place is playing with series, inequlities, and magic tricks. If you are a get-your-hands-dirty kind of mathematician, then you should never let this book leave your side.

Readability:
I think that it may be a different style than most people are used to, but once you get past that I think I would call the readability nearly perfect. He strips away most general useless commentary (for example, in Gallians poor algebra book, "In high school, students study polynomials with integer coefficients, rational coefficients, and perhaps even complex coefficients"). In Rudin, you get no nonsense -- only math.

The real trick to getting in his swing of things is to MAKE SURE YOU COMPLETE HIS PROOFS. They are extremely slick and often are polished in such a way that it's like his little secret. If you can't do one on your own, just ask the prof in office hours or put it aside for later. The proofs are not presented in this way as to imply that you should just accept them, he wants you to dig in and justify the intermediate steps for yourself, so do it and you'll be good by Ch. 3, I promise.

Exercises:
Many exercises in this book are often found as theorems in other books. What's so unique about this book is that very few problems are solved by simple definition pushing, especially as you go further into the book. That's why I call this the get-your-hands-dirty book, because you'll be forced to, and believe me you'll recognize changes in the way you think if you do this diligently. So, do as many exercises as you can, esp in Ch. 2 and Ch. 4, they will help you the most in this book. What's great about the problems is that they challange you to make REAL connections between ideas and create your own equivalent ways of thinking about the subject. I often have to conjecture and prove several lemmas to avoid wimping out and using "clearly" in my proofs.

Suggestions:
If you really really love math and know in your heart that you need to get better to be happy in life, you should cover Ch.1-Ch.6 before Juior year of college and finish it before grad school. I also suggest using this book as a stepping stone to more advanced books -- see Halmos' Measure Theory and know it before grad school.

Finally, DO NOT BE AFRAID! You really have to commit to this book before getting into it, do not be afraid. My best advice to any mathematician is to know your weaknesses, BUT to respond promptly to them.