This book is written in the best Mac Lane style, very clear and very well organized. It also benefits from Moerdijk's extensive work organizing the theory of Grothendieck toposes by elementary means. The reader should have basic graduate knowledge of algebra and topology. The book is long because it gives very explicit descriptions of many advanced topics--you can learn a great deal from this book that, before it was published, you could only learn by knowing researchers in the field.

This text is a very readable presentation of first year graduate abstract algebra. The material is presented with notations similar to that of Jacobsen in his "Basic Algebra" texts, and is useful as a review text for qualifiers, or for independent study.

After getting frustated by nearly all the so-called "authoritative" books on abstract algebra (Lang, Hungerford, Jacobson), I really can say that MacLane/Birkhoff is the best die-hard classic on algebra. Now I must stress that this book IS NOT out-of-print: the third edition is actually published by AMS/Chelsea.

There's an interesting thing about the evolution of this book: the first edition has become famous among mathematicians, because it brought for the first time an elementary exposition of categories and universal constructions, directly from the horse's mouth (MacLane founded the theory of categories together with S. Eilenberg; Birkhoff was the creator of the theory of lattices), which is used as a basic tool throughout the book; it also contained unusual topics such as multilinear algebra and affine and projective spaces, but no Galois theory. The second edition has gained a chapter on Galois theory, but has lost the part on affine and projective spaces.

The third edition is the best! It has recovered the part which was lost in the second edition, and had its exposition considerably polished. While most other books expose abstract algebra as a ugly, prawling monster, MacLane/Birkhoff manage to explain quite esoterical topics (many of them created and/or developed by themselves) in a surprisingly natural and tasty way (compare it with the dry, encyclopaedic style of Hungerford and Lang); although quite big, the book supports several ways of reading and teaching its parts without sacrificing clarity. Another great quality: it is INSPIRING, in the sense that it develops a powerful algebraic intuition, which is, in my opinion, the main obstacle one has to face to learn algebra.

This book is a fairly good introduction to the ideas of category theory by one of the creators of the field. Unfortunately, the book is sometimes sort of confusing, and doesn't give many as many examples as I would like. Category theory (while it has become a field in its own right), is really a way of thinking about mathematics. The way you learn a way of thinking is by working out examples & doing excersizes, but this book doesn't provide as many connections to other areas of math as it should.

I don't think that this book was really intended "for the working mathematician," but rather for someone with some independent interest in category theory.

This book is extraordinarily well written. It covers the necessary topics in a concise, orderly manner. HOWEVER, it presumes a substantial amount of knowledges concerning various algebraic/abstract structures in the field of mathematics. If you already have had experience with such structures, and are simply looking to understand them from a different perspective - this is the book for you. However, if you have limited knowledge with regards to advanced math (ie - grad level math) then try the book 'Arrows, Structures and Functors: The Categorical Imperative' by Manes and Arbib. This introduces the reader gradually to simple algebraic structures, monoids, groups, metric spaces, topological spaces, and the categories that can be built around them.

This book is a classic. Clearly written, drawing on a vast number of different applications and motivations for the subject. Eilenberg and Mac Lane created category theory and this book is alive with the very style of thought Mac Lane brought to it in the first place. It is obvious that Mac Lane wrote each page, and each exercise, with a view of the whole book in mind. He starts with the very basics, assuming indeed that you know nothing of category theory. He goes on to adjunctions, limits, the adjoint functor theorems, monads (triples), monoidal categories, Abelian cateories, Kan extensions, higher dimensional categories, and categorical foundations. It is a masterpiece and one of the great books in mathematics.

Modern algebra is an extraordinary topic and Birkhoff and MacLane do a superb job of exploring it. However, as is often the case with mathematical texts, the material can be somewhat dry.

I have just started reading this book, and already I am
enthralled by the beauty and elegance of the authors'
exposition. Assuming nothing more than an acquaintance with
school algebra and a little geometry, they develop
the basic properties of central algebraic structures, including
rings, groups and fields. These are treated by reference to
familiar examples, such as the ring of integers and the
rational, real and complex fields. Everything that one learned
in school algebra is to be found here, though, as is to be
expected, each topic is treated at a rigorous, mathematically
sophisticated level. In the first two chapters, the properties
of the integers and rational numbers are gradually examined,
ultimately down to the definition of addition and multiplication
on the basis of Peano postulates. The authors then consider
polynomials, the real and complex numbers, vector spaces, linear
algebra and other topics.
The writing style is clear, concise and elegant, with each new
concept being carefully defined as it is introduced. The proofs
achieve a satisfying balance between detail and brevity. Indeed,
reading the proofs and completing the exercises would do much, I
am sure, to enhance a reader's mathematical facility.

If you are interested in acquiring a deeper understanding of
algebra, this book should serve as an excellent introduction.