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Rating: 
Summary: Comprehensive...but "Bourbaki" in its style
Review: Algebra is a subject that must be mastered by anyone these days, although at varying levels. The applications of algebra permeate all fields of human endeavor, and for students, both at the elementary level and advanced graduate level, it can be a subject that is esoteric and on the surface removed from real world applications. This book, although comprehensive in its treatment of graduate level algebra, does not motivate the subject well, but instead treats it from an encyclopaedic, formal point of view. There are some exceptions to this, particularly in the chapter on Galois theory, as the author does a fine job of detailing this subject. If he would have approached the other chapters like he did in this one, this would be an excellent book. Unfortunately though too much of the book is without in-depth explanation, and it leaves the reader wanting.
Rating: 
Summary: Impressive, if not so easy to read
Review: As a reference, this is simply an amazing book; tons of information are crammed into this book. The flip side is that if you are seeing this information for the first time, the presentation can be a little daunting. I started using this book in a class last year and hated it at first, because the presentation of material here is very densely packed together and not written for maximum clarity. For example, the chapter on category theory was the first time I'd seen the subject, and I found it frustrating, unlike the presentation given by, say, Rotman in his Algebraic Topology text. All of that said, though, I appreciate the book more now looking back on the material. Overall, if you haven't seen the material before, this is a fine book as long as you've got someone to help you through the rough spots. As a reference, though, this book is extraordinary.
Rating: 
Summary: comprehensive and rigorous
Review: I agree with others, this book isn't for those who are
unmotivated. But if you know what you want from algebra
and need a comprehensive, rigorous treatment, this book is
great. I was able to learn on my own from it, and I'm not only
a non-math major, I had no access to any instructors. That
should tell you something. Aside from that, the book has a
few minor quirks, like exercises which aren't really doable
or exceedingly difficult (i know because I've seen these
questions answered in other books). But there are few of those
so it's a minor nuisance.
Rating:
Summary: Well worth the read for any budding mathematician
Review: I've been acquainted with several introductory graduate algebra
books over the years, and prefer this one for its coverage of all the fundamental areas (groups, modules, rings, linear algebra, fields, and category theory), being concise, and providing great care when outlining each proof. If one compare's the amount of material in this book to Jacobson's "Basic Algebra Vol 1", Grove's "Algebra", or Herstein's "Abstract Algebra", Hungerford's book gets the nod.
Moreover, I much more prefer the concise definition, example, theorem, proof format over the more colloquial approach, as can be found in Jacobson's text. For me at least, the payoff for reading an algebra text is the beauty found in the logic and reasoning from which very profound results arise from the complex interaction and use of more straightforward ones. And this is exactly where Hungerford's book shines through in tremendous glory. When outlining a proof he does an outstanding job in citating the results from previous Chapters that are used. For me this is the strength of algebra (In geometry I cringe when I get a picture for proof, and in analysis it is often quite complicated to verify that a given situation possesses the appropriate conditions needed to invoke some famous lemma or theorem). One last good word about this book: I found the exercises both in abundance (after each section) and quite reasonable for a first year grad. student. Happy reading.
Rating:
Summary: Deceptively Wonderful
Review: OK here's the truth: This book is an awful text when accompanied by not so great prof is teaching from it (e.g. one who delivers nothing but the text). BUT... once you begin to understand enough to know that the "trivial" "exercise" and "left to the reader" proofs are quite straightforward, the book is probably the best reference in Algebra you can hope for.
Rating:
Summary: The bee's knees
Review: This book is the Basic Language of Mathematics (by J. J. Schaffer) of the Algebra world. Without doubt it is an excellent dictionary of general facts about algebra. But learning by it will leave one with at best amusing memories and a nervous twitch. Just for a taste, "This proof has two parts. The first is easy. The second is left to the reader." About half the proofs in the book go like this. And so at the end of each section, the reader is left with just the dry theorems to attempt the exercises, without the slightest idea of how problems of a certain type are actually proven or even approached. And oh, the exercises. A few are easy. A few are open problems. The rest in between seem to at one point have been at the core of someone's respective masters thesis. This book has three genuinely good uses. If you have a doctorate in pure Mathematics, a respectable doctorate that has nothing to do with PDEs and the thesis for which took longer to write on paper then it did to format the pictures to fit the margins, and you want to look up how much of the ring structure of R is inherited by R[x] in under 3 minutes, then this book belongs on your shelf. If you have taken at least two algebra courses at the graduate level (Real graduate, not graduate equivalent. Most of my Algebra I class had two pretty good undergrad algebra classes coming in, and got slaughtered by Hungerford), then this book can make for a good review of basic algebra you should already know. Finally, if you are already comfortable with algebra but would like to know more about category theory, this book offers a different perspective on the subject that might be insightful, so long as you don't grow a hatred of the word 'free'.
Rating:
Summary: The bee's knees
Review: This book is the Basic Language of Mathematics (by J. J. Schaffer) of the Algebra world. Without doubt it is an excellent dictionary of general facts about algebra. But learning by it will leave one with at best amusing memories and a nervous twitch. Just for a taste, "This proof has two parts. The first is easy. The second is left to the reader." About half the proofs in the book go like this. And so at the end of each section, the reader is left with just the dry theorems to attempt the exercises, without the slightest idea of how problems of a certain type are actually proven or even approached. And oh, the exercises. A few are easy. A few are open problems. The rest in between seem to at one point have been at the core of someone's respective masters thesis. This book has three genuinely good uses. If you have a doctorate in pure Mathematics, a respectable doctorate that has nothing to do with PDEs and the thesis for which took longer to write on paper then it did to format the pictures to fit the margins, and you want to look up how much of the ring structure of R is inherited by R[x] in under 3 minutes, then this book belongs on your shelf. If you have taken at least two algebra courses at the graduate level (Real graduate, not graduate equivalent. Most of my Algebra I class had two pretty good undergrad algebra classes coming in, and got slaughtered by Hungerford), then this book can make for a good review of basic algebra you should already know. Finally, if you are already comfortable with algebra but would like to know more about category theory, this book offers a different perspective on the subject that might be insightful, so long as you don't grow a hatred of the word 'free'.
Rating:
Summary: Encyclopedic, but dry.
Review: This tome is probably the best single-volume REFERENCE for basic abstract algebra at the graduate level. It touches on almost every important subject. However, the style is very much in the way of an efficient, concise, statement of fact rather than a lucid expository of subject-matter. This is an excellent reference, but for the task of learning the material (especially if without a lecture), I would recommend Dummit and Foote or Steinbeck (the former for advanced undergraduate, the latter for purely graduate study). Also, while this is very comprehensive, it simply cannot fully treat everything in all subjects. For example, very little is given in the way of group (co)homology. For the specialist, you should instead invest in more specific books (e.g. Robinson).
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