Rating: 
Summary: A great reference, and a poor first year text
Review: After having struggled with this book for most of my first year algebra class I grew to hate it! I now find myself refering to this book time and time again. This book really has it all. If you already have a good background in algebra and want a great referance, this book is a must. If you are taking algebra for the first time I would avoid this book and look at another intro. to algebra text like Grove (one of my favorite math texts).
Rating:
Summary: A great reference, and a poor first year text
Review: After having struggled with this book for most of my first year algebra class I grew to hate it! I now find myself refering to this book time and time again. This book really has it all. If you already have a good background in algebra and want a great referance, this book is a must. If you are taking algebra for the first time I would avoid this book and look at another intro. to algebra text like Grove (one of my favorite math texts).
Rating:
Summary: If you don't know algebra, this is not for you.
Review: As always with Lang, this book is not user-friendly: it makes a pretty hard reading unless you already have a fair knowledge of algebra as covered by other good texts like Cohn (volumes 1 through 3). If you are looking for something as reference, you should turn to more comprehensive works like Bourbaki.
Rating: 
Summary: This is the standard reference for algebra
Review: Concise but comprehensive, Lang's book really has no peer as a reference text for algebra. Both Cohn's and Jacobson's books omit far too many topics. Jacobson's book, which I am more familiar with, follows a very linear structure, which I find limits its appeal both as a reference text and as bedtime reading. This book ought to be challenging for any undergraduate, and perhaps even for some graduate students, but much of the material is essential. The book, unfortunately, contains few examples. This is especially problematic in the section on homology theory, where the abstraction becomes nearly overwhealming. Working concrete examples such as Ext, Tor, or the (co)homology of groups into the text would have been helpful.
Rating: 
Summary: Good one volume expostion of basic algebra
Review: Difficult to agree with my learned friend from Jackson,Mississippi that the chapters on groups and rings were boring. I must congratulate him on finishing the book in one week. More seriously, the book provides enough coverage of commutative algebra,Galois theory and homological algebra as to enable one to tackle the books by Eisenbud and Hartshorne on commutative algebra and algebraic geometry respectively.There are rival treatments by Cohn and Jacobsen but Lang beats them for conciseness.Lang is notorious for errors and omissions in his books and so one would expect a reader to have considerable 'maturity', i.e. the ability to correct proofs or fill in missing details.
Rating:
Summary: NEEDS revision!
Review: I hope that in the coming reprint the author would take care to correct the numerous typos and errors in the 3rd edition. Though books are seldom free from mistakes, those in this book are particularly frustrating and confusing. The author did a lot of rearrangements from the 2nd to the 3rd edition, but many self-references in the book are not updated, and I could never figure out where the author refers to at some places. Otherwise, I agree with the general opinion that this book is good but often too terse.
Rating:
Summary: A worthwhile pain in the [behind]
Review: I must concur with my fellow readers that in fact Langs Algebra text is extremely dry, the examples are sparse (as compared with, say, Hungerfords Graduate text), readers are left to fill in the gaps which exist within the majority of proofs and, finally, about the exercises; for the most part the exercises abound, they are challenging, non-trivial and in general are extensions of the material, which for whatever reason, have been relegated to the status of mere exercise. But for those who have a 'Solid' foundation in Algebra, preferably at the level of a Junior-Senior undergraduate who has completed courses in Linear Algebra, Modern/Abstract Algebra, then this text is worth its weight in gold. For those individuals who have either chosen to make Mathematics their career or those who are Mathematically gifted, a text of this stature must be appreciated for exactly those reasons I used to 'negatively' criticize this text. For example, when doing research at any level above that of advanced undergraduate, the researcher should have the confidence, temperance, skill and desire to fill in missing gaps within proofs since the ability to do so is an excellent gauge of how well one actually understands the given material. It would seem to logically follow from this that the researcher would then benefit from choosing a text that contained exercises, which were not trivial calculations or the requirement of proving somthing that is either routine or standard. Instead, major rewards, in the form of confidence and a deeper understanding, are a result of struggling through difficult problems and, in general, problems which lead you toward self-discovery, i.e. those which are extensions of the given material. For these reasons I highly recommend this text to all members of the Mathematical community who desire more bang for their buck since this will serve them well, both as a text for further study and as a lifelong reference.
Rating:
Summary: great text, requires mathematical maturity.
Review: I purchased this text after completing my first course in abstract algebra. I took the class (MA407) through my high school (via a college release program) in 12th grade. For that class we used Gallian's text "Contemporary abstract algebra." I was not fully satisfied with the text's applied flavor so I used Artin's and Herstein's texts as supplements. Now I realize that my purchase of Lang's text was probably premature. It is a difficult text, and although I enjoy it, getting through any given chapter takes me a long time. I have not completed the text (as a matter of fact I am still in section I), but I feel that I can see the basic style of the text. Lang's text is well written (although I keep running into small errors), and probably should be in the library of any graduate student in the pure mathematics. As for me, I think I jumped at this book too soon and should go back to Artin and complete that text first. Lang's text is probably better suited for a second class in algebra at the graduate level (there are some other advanced tests such as Jacobson's, but I believe that Lang's text is the only well written text book in algebra which covers such a wide range of topics in proper detail). In fall 1999, my freshman year, I will take graduate algebra and graduate real analysis, I hope after those classes I will be able to tackle Lang's text in a more reasonable manner.
Rating:
Summary: Excellent if you have the requisite mathematical maturity
Review: I sometimes joke that "mathematical maturity" is the ability to understand poor exposition. Lang's proofs are often too terse, and even experienced readers will sometimes have to work hard to fill in all the gaps. For this reason this book is not the best choice for most beginning graduate students. Nevertheless, time and time again in my study of algebraic number theory and algebraic geometry, when there has been some nugget of algebra that I had forgotten or never learned, I have found it in Lang and not in other standard texts. So for me, this book is an indispensable reference. Lang also has a knack for giving insightful summaries of advanced topics. Most other authors will at most mention an advanced topic without really telling you anything about it, but Lang actually gives useful introductions to a large number of topics of current research interest.
Rating:
Summary: Excellent if you have the requisite mathematical maturity
Review: I sometimes joke that "mathematical maturity" is the ability to understand poor exposition. Lang's proofs are often too terse, and even experienced readers will sometimes have to work hard to fill in all the gaps. For this reason this book is not the best choice for most beginning graduate students. Nevertheless, time and time again in my study of algebraic number theory and algebraic geometry, when there has been some nugget of algebra that I had forgotten or never learned, I have found it in Lang and not in other standard texts. So for me, this book is an indispensable reference. Lang also has a knack for giving insightful summaries of advanced topics. Most other authors will at most mention an advanced topic without really telling you anything about it, but Lang actually gives useful introductions to a large number of topics of current research interest.
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