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Advanced Modern Algebra

Prentice Hall
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This is a genuine grown-up abstract algebra book. It is not meant for a first abstract algebra course (nor perhaps even a second). However, it is, wisely, self-contained, and so as the author says in his Preface, "To accommodate readers having different backgrounds [meaning different topics in their undergraduate abstract algebra courses], the first three chapters [titled "Things Past", "Groups I", and "Commutative Rings I"] contain many familiar results, with many proofs merely sketched."

Nonetheless, fortunately, the author seems pretty much incapable of being sketchy, or un-appreciative of "elementary" topics. For example, on page 55, explaining that not all binary operations are associative: "A binary operation allows us to multiply two elements at a time; how do we multiply three elements? There is a choice..." And, on page 88, "The Third Isomophism Theorem is easy to remember. In the fraction (G/K) / (H/K), the K's can be cancelled."

Rotman holds to this practice throughout his book, not only in the first three "easy" chapters but also in the other chapters [Fields, Groups II,Commutative Rings II, Modules and Categories, Algebras, Advanced Linear Algebra, Homology, and Commutative Rings III]. His development and proofs are complete and, usually, clear; he doesn't worry about wasting words, or about giving his own spin on things, or getting pedagogical. For example, p. 402, after definiing the degree-lexicographic order on finite products of the ordered set of natural numbers: "First check weights... then, if there is a tie... order them lexicographically" and, on the same page, "the next proposition shows, with respect to a monomial order, that polynomials in several variables behave like polynomials in a single variable." And page 442, introducing categories: "Imagine a set theory whose primitive terms, instead of set and element, are set and function. How could we define bijection, cartesian product, union, and intersection? Category theory will force us to think in this way." And page 589, describing "diagram chasing", "we select an element and, at each step, there is essentially only one thing to do with it." And page 795, as he takes on the task on characterizing extensions of a given abelian group K by a given group Q: "We can regard a factor set as the obstruction to a lifting being a homomorphism; that is, factor sets describe how an extension differs from being a split extention."

Indeed, while not specifically "user-friendly", this book is far from unfriendly; for example, the author has no qualms about writing, occasionally, in the first person. Page 279: "The Zassenhaus Lemma is sometimes called the butterfly Lemma because of the following picture. I confess that I have never liked this picture; it doesn't remind me of a butterfly and it doesn't help me understand or remember the proof." And p. 781: "When I was a graduate student, homological algebra was an unpopular subject. The general attitude was that it was a grotesque formalism, boring to learn, and not very useful once one had learned it... This attitude changed dramatically when J.-P. Serre characterized regular local rings using homological algebra..."

I view this stance almost as a necessary thing, in that, not only does it add a relaxed, human, almost literary touch, and not only does it convey so much of the spirit and flavor of math itself, but it also adds mathematical insight. In one of my "math poems" I wrote, "Beauty isn't as pretty without truth." Now I'm saying "Truth isn't as true without beauty."

This book is impressive, and I feel grateful to have had the opportunity to have read it. (Usually, when I'm asked to review a book with the words "abstract [or modern] algebra" in the title, my response is "O goodie!" But this time I got a huger bargain than I expected.) The author, in the Preface, describes his book — all 1012 pages — as a "reference", and he's right. But, he adds, it's "not only an appretizer, but a hearty meal as well."

However, if it's going to serve as a reference, it could use a better index; likewise better Special Notation on p. xiii. In the former, I searched mostly unsuccessively for the first appearances of Ext and Tor and of "universal mapping property". And in the latter, where is coker f (to match up with ker f) and indeed, where are Ext and Tor? I believe that both of these "references within this reference" could have been made more complete without being overbearing.

Here is one further suggestion, with respect to clarity: dual concepts could be treated more "dual-ly". For example, compare page. 474 (defining projective modules) with pages 480-1 (defining injective modules). Also, I would turn the diagram for the definition of "projective" ninety degrees; liftings could easily be made to look more like liftings! A wording to go along with the definition might also be appreciated by a reader new to this stuff — and, again, add more mathematical insight. (Perhaps something like "Any map from P to a quotient module can be lifted." And then the dual explanation for injective modules: "Any map to E from a submodule can be extended.")

At the risk of being nit-picky, here are a few passages which, if it were my book, I would do differently: P. 86: "Any two infinite cyclic groups are isomorphic to Z" must have been an oversight. (Two taken together how?! I'm sure he meant to write "Any infinite cyclic group is isomorphic to Z.") And on page 392, wouldn't it be clearer and shorter to state parts (i) and (ii) of the hypothesis "collectively" as: J contained in T contained in root-J (that is, T intermediary to J and root-J)? And on page 814, Proposition 10.32 (that every module has a free (and hence projective) resolution provides the first appearance in the book of a long exact sequence, and therefore I feel that the actual long exact sequnce should be displayed, as well perhaps as more attention given to the proving that it's exact. (It's only an extra line or so.) And on page 832 (The Comparison Theorem), I would try to make the statement more clear, perhaps with two diagrams rather than only one; the way it is, it's a little difficult to realize that only the map f is given, not the various fn. (That is, I'd try to visually distinguish hypothesis from conclusion.)

Also, for what it's worth, there seems to be a typo on page 864; in Proposition 10.96, A should, I believe, be F.

It's actually quite impressive that, among 1012 pages, I found so few nits to pick! In a book with so much quantity (as well as quality), it is not surprising that there are a great many topics which (in my experience) are rare in algebra texts (even at the graduate level). These topics are too numerous to mention, but some of my favorites are: a complete proof that any subgroup of a free group is free, extensions of the division and Euclidean algorithms for polynomials of several variables, exterior algebra, a complete treatment of semi-direct products, and gobs and gobs of long exact sequences. Very seldom does he use the phrase "too long to present here" (although he sometimes does refer us to other texts — sometimes his own — for further results and proofs, making us feel very humble indeed!).

Marion Cohen teaches at the University of Pennsylvania.


Date Received: 
Saturday, January 1, 2005
Include In BLL Rating: 
Joseph J. Rotman
Publication Date: 
Marion Cohen
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Special Notation.

1. Things Past.

Some Number Theory. Roots of Unity. Some Set Theory.

2. Groups I.

Introduction. Permutations. Groups. Lagrange's Theorem. Homomorphisms. Quotient Groups. Group Actions.

3. Commutative Rings I.

Introduction. First Properties. Polynomials. Greatest Common Divisors. Homomorphisms. Euclidean Rings. Linear Algebra. Quotient Rings and Finite Fields.

4. Fields.

Insolvability of the Quintic. Fundamental Theorem of Galois Theory.

5. Groups II.

Finite Abelian Groups. The Sylow Theorems. The Jordan-H”lder Theorem. Projective Unimodular Groups. Presentations. The Neilsen-Schreier Theorem.

6. Commutative Rings II.

Prime Ideals and Maximal Ideals. Unique Factorization Domains. Noetherian Rings. Applications of Zorn's Lemma. Varieties. Gr”bner Bases.

7. Modules and Categories.

Modules. Categories. Functors. Free Modules, Projectives, and Injectives. Limits.

8. Algebras.

Noncommutative Rings. Chain Conditions. Semisimple Rings. Tensor Products. Characters. Theorems of Burnside and Frobenius.

9. Advanced Linear Algebra.

Modules over PIDs. Rational Canonical Forms. Jordan Canonical Forms. Smith Normal Forms. Bilinear Forms. Graded Algebras. Division Algebras. Exterior Algebra. Determinants. Lie Algebras.

10. Homology.

Introduction. Semidirect Products. General Extensions and Cohomology. Homology Functors. Derviced Functors. Ext and Tor. Cohomology of Groups. Crossed Products. Introduction to Spectral Sequences.

11. Commutative Rings III.

Local and Global. Dedekind Rings. Global Dimension. Regular Local Rings.

Appendix A: The Axiom of Choice and Zorn's Lemma.


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Monday, July 18, 2011