(I am writing about the 2nd edition, which I used as an undergraduate.)

This book is intended for a one semester senior-level honors course at a reasonably good undergraduate institution, for which it is perfect. Students who are less interested in pure mathematics or are somewhat weaker should go to Gallian's book, which is also excellent. Students who are weaker still maybe should seek out Fraleigh.

Other reviewers are correct about the group theory being the strength of this book; ring and field theory are OK but short, but remember that this book is intended for a one semester undergraduate course. (Herstein was a ring theorist. It is natural to speculate that he chose the topics he did because of the course, not because of personal interest...) The optional topics (simplicity of A_n, Liouville's Criterion, etc.) are excellent.

"Topics in algebra" is supposed to be a year-long version of this book. That one is sometimes called "Herstein" and this one is "Baby Herstein". Happily though, Baby Herstein still has content, unlike "Baby Hungerford"...

Abstract algebra (AKA "algebraic structures", "modern algebra", or simply "algebra") can be a difficult topic depending on its presentation. The difficulty comes in the abstractness of the topic (generalizations that give us useful properties), not the complexity of the area (though, further study can provide some of this).

Although the several texts I have seen are useful in their own right, I don't believe there's a better text for beginners (or, perhaps, to strengthen shady concepts for further courses) on the subject. Herstein presents concrete examples before proving abstract concepts (something students who have only had courses on the several calculus, discrete math, probability, and matrix theory will find invaluable).

The text is clear and concise. The length is short without omitting any pertinent ideas (other books tend to spend a wealth of pages on anomalies -- which can be good...but then we could really make volumes on the subject). The book starts with a basic (but complete) introduction to sets, groups, symmetric groups, rings, fields and ends with some special topics (simplicity of A4, finite fields existence and uniqueness, cyclotomic polynomials, Liouville's criterion, irrationality of pi). As with most texts on the subject, there are no solutions provided.

The tone taken in the work caters to the student who has not had a course in abstract theory and proofs (ie- courses in analysis or topology ; a number theory course, in my experience, is not rigorous enough for the average student supply the necessary background).

For more difficult and robust presentations, look towards Artin's "Algebra" or Hungerford's "Algebra" (the latter being part of the Graduate Texts series).

For a more application (example-based) text (usually simpler for the less advanced student), look towards Gallian "Contemporary Abstract Algebra." This text is one of the rare occasions where the odd-numbered problems have solutions (but if that's all you're looking for, go to some of the Schaum's series). It is a bit basic (spending most of its examples on more familiar concepts), but also hits on some historical notes and examples that are good conversation pieces (something more mathematicians could use).

Also, as a sidenote, the editions have not changed the content at all. I would suggest getting an older edition...

We are so lucky that Herstein, as a great write, wrote this text in the last two years of his life. I think I am careful enough to say this is the best text I ever seen for years. Certainly, "best text" to different people are different. For every student who wants to be a mathematician, I stronly recommend this as the first text on abstract algebra. If you have a month, reading and doing all the problems, you will love mathematics.

And I want to point out a mistake by another reviewer here. He said something about 30 years. However, this book is written 12 years ago.