Advanced Modern Algebra [Hardcover]

Joseph J. Rotman (Author)

Book Description

ISBN-10: 0130878685 | ISBN-13: 978-0130878687 | Publication Date: May 5, 2002

This book's organizing principle is the interplay between groups and rings, where “rings” includes the ideas of modules. It contains basic definitions, complete and clear theorems (the first with brief sketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of definition-theorem-proofs, this text put results and ideas in context so that students can appreciate why a certain topic is being studied, and where definitions originate. Chapter topics include groups; commutative rings; modules; principal ideal domains; algebras; cohomology and representations; and homological algebra. For individuals interested in a self-study guide to learning advanced algebra and its related topics.

Editorial Reviews

Review

"... a highly welcome enhancement to the existing textbook literature in the field of algebra." -- --Zentralblatt fur Mathematik

[T]his is an excellent book containing much more than what is likely to be covered in a standard graduate course. [In this new edition] several topics have been added…The index …[has] been much improved. … The best features of the first edition are retained. --Fernando Q. Gouvêa, MAA Online --This text refers to an alternate Hardcover edition.

From the Back Cover

This book's organizing principle is the interplay between groups and rings, where “rings” includes the ideas of modules. It contains basic definitions, complete and clear theorems (the first with brief sketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of definition-theorem-proofs, this text put results and ideas in context so that students can appreciate why a certain topic is being studied, and where definitions originate. Chapter topics include groups; commutative rings; modules; principal ideal domains; algebras; cohomology and representations; and homological algebra. For individuals interested in a self-study guide to learning advanced algebra and its related topics.

See all Editorial Reviews

Product Details

• Hardcover: 1040 pages
• Publisher: Prentice Hall (May 5, 2002)
• Language: English
• ISBN-10: 0130878685
• ISBN-13: 978-0130878687
• Product Dimensions: 9.2 x 7.2 x 1.6 inches
• Shipping Weight: 3.7 pounds
• Average Customer Review: 3.6 out of 5 stars  See all reviews (7 customer reviews)
• Amazon Best Sellers Rank: #609,357 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

46 of 47 people found the following review helpful:

4.0 out of 5 stars Good for Self-Study, August 21, 2003

By A Customer

This review is from: Advanced Modern Algebra (Hardcover)

This is a tough book to review, because it is not clear who the real audience is supposed to be. The author says that it is aimed at first-year graduate students, with a bunch of extra material that can be referred back to during the second year and beyond. The earlier chapters also include efficient reviews (with sketched proofs) of material that should be familiar to those who have taken undergraduate algebra.

This characterization is debatable. Based on my experience reading most of the first six chapters (the first 400 out of about 1000 pages), I would say that the level of sophistication is roughly that of Dummit and Foote's "Abstract Algebra", which is usually considered an undergraduate book. D&F can sometimes be harder to read, and that is in part because Rotman's exposition is better (in my opinion), but also because D&F introduce more difficult material earlier. Whether D&F's approach is better is questionable; I find Rotman to be a much smoother read, but the organization is quite different -- for example, one does not encounter noncommutative rings until deep into the book, whereas Dummit and Foote introduce them immediately upon defining rings. On the other hand, early in the coverage of D&F's chapter on rings, one has to digest Zorn's Lemma and its applications almost from the beginning, whereas Rotman (I think wisely) pushes this back into a later section. In general, D&F introduce a lot of hairy examples that by themselves require a lot of effort to digest (thereby impeding the reader's progress through the core material), whereas Rotman's examples tend to be straightforward, at least as new concepts are being presented.

So, overall, the exposition flows more smoothly in Rotman's book, and the reader can cover the basics more quickly with less time spent on tangential examples and early generalizations. Also, Rotman's proofs are usually much cleaner and the overall style is very nice. It's more pleasant to read than Dummit and Foote. But this comes at a cost: Dummit and Foote do cover more material, and generalize at an earlier stage, than Rotman does.

But my biggest gripe concerns the exercises. Put simply, Rotman's are far too easy for what is being pitched as a graduate course. In fact, they are in general far easier than the homework problems I sweated through when I took honors undergraduate algebra. They're barely adequate to convince the reader that he has a basic grasp on the material, and there are almost no hard ones, let alone really tough, thought-provoking open-ended problems like one encounters in Herstein's "Topics in Algebra" (an undergraduate book). There are certainly no exercises in Rotman's book that would be of any use for a graduate student preparing for qualifying exams. They're not even much of a workout for a decent (honors student) undergraduate.

So, what is this book good for? I think it's great for reading material that is usually harder to understand elsewhere. Rotman has a real knack for clear mathematical exposition, and some of the chapters are a real joy to read. (Side note: there are also a lot of typos, at least in the first printing. The author maintains an errata list at his web site, and a second printing is coming soon. There are still many errata that he didn't catch, but they're fairly minor and do not detract significantly from the reading.) But this is simply not suitable for a primary graduate text or reference. Most good schools are going to demand more of their graduate students, and one is inevitably going to have to read Lang or Hungerford (and work through their exercises) to achieve competence at the graduate level. Rotman's book is a kinder, gentler book upon which to fall back when those books are inscrutable, as is all too common. I do recommend it highly for that purpose -- I think it's a very good secondary book.

20 of 22 people found the following review helpful:

5.0 out of 5 stars An excellent Text, July 18, 2003

By 

Jason Schorn "j schorn" (Spokane, WA) - See all my reviews
(REAL NAME)   

This review is from: Advanced Modern Algebra (Hardcover)

To begin with, don't let the title scare you. After having read through Rotman's book I am suprised that this text had not crossed my path earlier. It is a wonderful book and must have for any inspiring Algebraist. Moreover, I am quite shocked that the larger universities have not adopted this book.

(a) This book could quite easily be used as the standard third/fourth year undergraduate introduction to Abstract Algebra. In particular, the first four chapters provide a solid foundation for a moderate paced one semester course at which point the instructor has many different options for additional topics based on the performance of his/her class.

(b) Those students that move on to the graduate level, and obviously to a university using this book, would both be familiar with the temperment and flow of the author as well as devoid of the requirement of having to purchase another expensive Mathematics text. For example, my undergraduate Algebra text was Hungerford's and post completion the logical step, being familiar with his style, was to purchase Hungerford's graduate text. For those not familiar, let me tell you there is a night and day difference with repsect to how the material is presented.

(c) The remaining 7 chapters take the willing student on a pleasant tour of ring/module theory, some advanced group theory (for the inspiring group theorist I highly recommend the authors graduate text "Group Theory"), algebras(linear included), Homology(some cohomology) and finally some algebraic number theoretic concept under the heading of Commutative Rings III.

(d) Lastly, Rotamn does not get needlessly bogged down in any one section of the book. The flow is smooth, to the point with precise definitions, examples, and ample exercises.

I have only two negative remarks: one, the failure to include more aspects of field/Galois theory. This may be due to the author already having published a book entitled "Galois Theory". Two, the failure to devote an entire section to Finite Fileds and possibly some its applications. But this failure is minimal since, at present, the majority of Algebra texts, fail to adequately introduce and motivate Finite Fields.

19 of 25 people found the following review helpful:

5.0 out of 5 stars Great Book!..., October 6, 2002

By A Customer

This review is from: Advanced Modern Algebra (Hardcover)

I previously purchased Rotman's First Course in Abstract Algebra, and fell in love with it. So when I saw he a Second Abstract Algebra book, I had to have it. I am currently taking a Graduate Level Modern Algebra course, and I find this book to be a great help in my Studies. I wouldn't be as interested in Modern Algebra as I am now if it weren't for this book. I love this book and I would reccomend it to anyone who is interested in Modern Algebra, or taking a course in Modern or Abstract Algebra.