Algebra

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Springer Science & Business Media, Jun 21, 2005 - Mathematics - 914 pages

"Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books."  NOTICES OF THE AMS

"The author has an impressive knack for presenting the important and interesting ideas of algebra in just the right way, and he never gets bogged down in the dry formalism which pervades some parts of algebra." MATHEMATICAL REVIEWS

This book is intended as a basic text for a one-year course in algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra.  For the revised third edition, the author has added exercises and made numerous corrections to the text.

 

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very special book .ha............

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Full disclosure: I am an undergraduate student teaching myself from Lang's 'Algebra' and I hope my review will help others who may be looking to do the same.
On the book, in general:
Lang has some idiosyncrasies that build up over the course of this tome—namely, differences in terminology— but it is well worth having. His categorical approach is indispensable, you will have to learn it at some point if your mathematical pursuits point you in this direction. If you have the patience, time and diligence, it is well worth your while to work out the examples and problems in the text as they will significantly broaden your understanding and perception of the subject in a manner that other 'standard' texts like Dummit & Foote would not.
My personal experience:
On a more personal level, my complaints with this text are limited. Lang faces the classic dilemma of 'quality vs. quantity' and has decided on the latter, in that many of the 'proofs' in this text, while full and acceptable, are not 'fleshed out' in the interest of making the book <1600 pages long! So, I would not recommend this book as a starting 'kit' to teach oneself abstract algebra except for, perhaps, the most talented students. I am currently undertaking that task and I find it exceptionally challenging, and I am by no means amongst the most talented students. There are some things that Lang leaves to the reader that are not all that trivial. The reader will have to connect the dots to figure out what Lang means on many proofs and will be forced to draw on their knowledge from the preceeding sections to do so.
As an example, Lang may state that a particular morphism is injective and will give a little blurb on his reasoning that is more-or-less a hint to reader on how to attack the issue of demonstrating his statement. Many of these sorts of statements, he presumes, will be obvious, and in many cases they are easily divined, but occasionally I find they are not. I believe this detracts from the overall quality of the book as a tool to teach yourself, but this should not be misconstrued as an attack on the book's worth. The text is obviously intended for a mathematically mature audience.
Lang's 'Algebra' is called a dry text. I can only comment on this by saying that if you are looking for an approach that actively attempts to engage readers, this is not it. However, the interested and self-motivated reader should have no problem dealing with this, I certainly have never found it a problem. If you are confident in your abilities and willing to consult outside texts for help, I would say to use Lang's book as your primary resource. Perhaps you are talented enough to muscle your way through Lang alone! If you are less confident, I would recommend you do not pick up Lang's tome as your primary resource! Take your time with other books and return to Lang afterwards for a rewarding categorical and homological approach that is the basis of modern algebra!
 

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Contents

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IV
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VI
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VII
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VIII
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XCV
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XCVI
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C
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XC
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XCI
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CIX
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CLI
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CLII
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CLIV
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CLV
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CLXXX
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CLXXXIII
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CLXXXIV
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CLXXXV
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CXC
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