15
Jul 15

"Monoids,R-Modules And Nonassociative Rings-These Are Some of My Favorite Things: An Updated and Expanded Suggested Reading List For Honors Undergraduate and Graduate Algebra Part I of II”

Once again, yello. Actually orange is the original font color on my Microsoft Word draft. It’s the font color I wanted to use originally at this blog-but people started sending me messages it was making them go blind. 

As Archie Bunker famously said by mangling the French in classically ignorant, working class American Conservative manner. kay sa-roo, sa-roo.

This post is an update of the last post I wrote here in 2010 before going into the twilight zone for the last few years. Many of you will notice a lot of overlap between the two posts-but you should notice a lot of differences, too. In any event, it’s my show, so pooh pooh if you don’t approve………………


For those who don’t know, I’m sort of an unofficial bibliophile for mathematical education. I inherited this love of textbooks and monographs from my inspiration, friend and unofficial mentor, Nick Metas. I was 18 years old when out of simple curiosity I called him in his office to ask him for direction in independent studies of mathematics beyond calculus-and he went on for 4 hours, naming just about every textbook and describing the subject of mathematics. That long-ago conversation is what started me on the path to becoming a mathematician.

Nowadays, the influence of Nick is very clear in my life: I have an extensive library of textbooks and monographs, people ask me all the time for references on subjects. I used to review books for the Mathematical Association Of America’s website before I couldn’t pay dues anymore. (I hope to begin doing that again at some point when I rejoin.) Like everything else, I have an opinion on most commonly used texts and monographs for all subjects-and I’m reading more every year. My hope is to begin my own small publishing company through my website by late this year. But that’s for the future.


I’ve been asked many times over the years to compose a master list of my favorite textbooks and/or monographs. On my spare time, I’ve been arrogant enough to do that in bits and pieces. Many of my posts at The Math Stack Exchange have taken this form, besides the aforementioned MAA reviews, of course.

What makes my opinions on references for mathematics different from everyone else and their mother’s lists of mathematics books is my background and life experiences. This has lead me to evaluate the quality of textbooks based on 2 criteria. Firstly, I look at mathematics textbooks from the standpoint of students, not researchers. I ask myself not which books will be the best presentation for researchers, but for talented young people aspiring not only to be researchers themselves someday, but to be educators teaching the next generation after them and presenting the material the way they wish it had been presented to them. The second concern of mine when I evaluate a book is what is the background of its intended audience? In mathematical higher education probably more than any other subject, one size most definitely does not fit all. A suitable undergraduate real analysis text at MIT will not serve well the average mathematics major at most small liberal arts colleges-just as one that will serve the average student at one of these universities will bore the hell out of their honors students who had the misfortune of going there instead of to a top flight school for any number of a hundred reasons. Every student is different and has different levels of preparation-and this does not mean they lack talent. This is a dangerous myth that tends to be perpetuated by those fortunate or wealthy enough to go to top schools. I’ll return to this point momentarily.

The list will probably undergo many revisions and additions before it reaches final form-but more importantly, I’ve decided to compose it in modular form i.e in components. This way, it’s broken into bite-sized components of manageable length that I can post here. It seems to me if I wait and try to compose it all at once-well, I’ll end up writing a 2,500 page book from the old age home I’ll be dying of cancer in. So let’s get started and hope that what little insights I can give can help neophyte students looking to broaden their knowledge base in subfields of math or are just looking for a little help in coursework they’re struggling in. Comments, input and suggestions are, of course, very welcome.

The first module here is my favorite subject in all of mathematics: algebra. (A ludicrous but sadly mandatory clarification: When a mathematics student or mathematician says ‘algebra’; it’s supposed to be understood he or she means linear and/or abstract algebra. High school algebra is, of course, the simplest special case of this wondrous arena. )

How do we define abstract algebra? Like most branches of modern mathematics, attempting a simple nonmathematical definition for non-mathematicians is a nearly impossible Catch-22 since it requires mathematical concepts to even attempt a meaningful definition. Entire philosophical treatises could probably be written attempting to answer the question and would probably fail. But I think we can try for a reasonable working definition here.

I think the best way to define algebra is that it is the general study of structures in mathematics. By a structure, we mean some kind of set -by which we mean naively a collection of objects-and a function f closed on S (the range of f is a subset of S) with a specified list of properties that characterizes that structure. For example, a group is a nonempty set S with a binary operation f such that f is associative, there is a unique element e in S such that for all elements a in S, f(e,a) = f(a,e)= a and for every a in S, there’s a unique a* such that f(a, a*)= f(a*,a)=e. Algebra deals specifically with these kinds of objects.

The pervasiveness of algebra in modern mathematics in the 21st century is astonishing. It’s more than the sheer scope of algebra itself, but the fact that most of the active areas of mathematics would not even exist without it. And I’m not talking about high-tech fields where algebra’s role is obvious-like deformation theory and higher category theory. I’m referring to the fact that most areas of mathematics are formulated in the 21st century in terms of algebraic structures. To give just one possible example of a legion, modern differential geometry would be unthinkable without the language of vector spaces and R-modules. Without tangent spaces and their associated local isomorphisms, it would be impossible to generalize calculus beyond Euclidean space. It would also be impossible to precisely define differential forms, without which most of the most interesting developments of manifold theory fall to dust. As a result, a student that’s weak in algebra needs to seriously reassess a career in mathematics.

So the least I can do is give my 2 cents on the current crop of books available.

The actual direct impetus for me writing up and posting this list was Melvyn Nathanson teaching the first semester of the year-long graduate algebra sequence at the City University Of New York Graduate Center in 2010. I began that semester sitting in on his lectures in order to begin preparations for the algebra half of my oral qualifiers for the Master’s Degree in pure mathematics at Queens College. Unfortunately, a combination of personal and financial issues prevented me from attending regularly. So that was the end of that. ( Dr. Nathanson’s lectures-and my occasional private conversations with him-are 2 of the things I miss the most about hanging out at the Graduate Center. I don’t know if he’s still active there. I’ll find out soon enough upon my return. )

I found Dr. Nathanson’s (he never told me it’s ok to call him Melvyn , so I’m going to be extra cautious as not to offend him) comments on the subject very interesting, as he has his own unique take on just about any subject. As proof, I offer this excerpt from the course’s syllabus:


In 1931, B. L. van der Waerden published the first edition of Moderne Algebra, two classic volumes, written in German, that were based in part on lectures by Emil Artin and Emmy Noether and that became the canonical work in abstract
algebra." The second edition appeared in 1937, and an English version, Modern Algebra, translated by Fred Blum and Theodore J. Benac, was published in the United States in 1949 and 1950. I and many other American mathematicians
learned algebra from the original English edition of van der Waerden. It is still a great work and I strongly recommend it for intensive study. The first volume of the seventh German edition of van der Waerden is also available in English translation, but I prefer the original. Van der Waerden's algebra begins with introductions to different algebraic structures. The first seven chapters are “Numbers and Sets," “Groups," “Rings and Fields, "Polynomials"“Theory of Fields," Continuation of Group Theory," and The Galois Theory." As proof of van der Waerden's influence, this continues to be the starting sequence of topics in most algebra courses and most algebra books, including the contemporary classic, Serge Lang's Algebra, which I also recommend. This course is different, not just in the sequence of topics, but in its philosophy. It emphasizes themes in algebra: Divisibility, dimension, decomposition, and duality,
and the course enables algebraic understanding and technique by developing these themes. The book includes all of the theorems expected in a graduate algebra course, but in a nontraditional order. The book also includes some important
topics that do not appear in van der Waerden or Lang.”

My perceived implication from the preface and his subsequent remarks was that Professor Nathanson hoped to eventually expand these notes into a textbook for a graduate algebra course. I don’t know if he ever followed through on this or what stage the book is at if he did.

But his comments got me thinking about the current state of algebra courses and the textbooks that form the basis of them. Nathanson’s experiences are not unlike those of most mathematicians of his generation: van der Waerden’s classic was the source from which he learned his algebra. Later mathematicians; particularly algebracists-such as my undergraduate algebra teacher, Kenneth Kramer-learned algebra from the earlier editions of Lang’s tome. (In fact, it was more personal for Kramer. As an honors undergraduate at Columbia in the late 1960’s, he was a student in the graduate algebra course taught by Lang himself-whose resulting lecture notes ultimately evolved into the classic text.) Most of the better universities’ graduate programs adopted Lang as the gold standard of first year graduate algebra, for better or worse, after the 1960’s. With a very few exceptions, this was the story until after the turn of the 21st century, when a host of graduate algebra texts came onto the market within a 5 year period. What was once a very sparse set of choices for this course is now a wide field of markedly diverse texts, many authored by very eminent mathematicians.

What follows is my attempt to form an amateur’s guide to these texts and my corresponding brief commentary to each. As a reviewer of textbooks, it seemed under the circumstances, that providing such a list to my erstwhile classmates in Nathanson’s course-as well as the mathematical world in general-would be a very positive undertaking. I don’t know if it would be wise, merely positive. I must add the disclaimer that I am by no means an expert; I’m merely a serious graduate student. Therefore, this reading list must be taken with a salt lick of caution as coming from an amateur and as such, it is seriously subject to revision as my knowledge grows and my mathematical style tastes change.

A major motivation in the evaluation of each of these books has been student-friendliness. Let me clarify greatly what I mean by that. A lot of top-notch mathematicians and students have an elitist, almost snobbish reaction to a textbook when you say its’ friendly. “Oh,you mean it spoon feeds the material to the brainless monkeys that pass for mathematics majors at your pathetic university? How amusing. Here at Superior U, we use only the authentic mathematics texts. Rudin.Artin Hoffman and Kunze. Alfhors. We propagate the True Word. Math is supposed to a struggle for those truly gifted enough to be worthy of it.“

Or something equally narcassistically pretentious.

I have a lot to say on this and related issues-but if I started going in depth about it here, I’d write an online book here. In future installments, I’ll begin to outline them in detail.

But in plain English, this is a bunch of crap.

The reason a lot of those “classic” texts are difficult to read isn’t because their authors were first-rate mathematicians and as such, their lessons are beyond the reach of mere mortals. In a lot of cases, it was simply because most of them never really thought about teaching; of being able to organize their deep understanding of their chosen fields -and as a result, they were very poor communicators. This lack of communication skill is reflected not only in their poor reputations as teachers, so often inversely proportional to their reps as researchers-but also in the resulting textbooks. Why don’t they? Well, again, it’s too complicated to fully go into here. But I will say that part of the reason, as any research mathematician of any prominence will tell you-is that they don’t get paid the big bucks and get the fancy titles based on how well students learn from them.

The sad part is that this myth has been perpetuated by the canonization of certain textbooks as The Books for certain classes, despite the fact that most students almost overwhelmingly despise them. And the reason why is simple: They just aren’t clear and well-organized. That makes the very act of reading them unpleasant, let alone actually learning from them. For the serious mathematics major or graduate student, this makes studying from such books virtually an act of psychic self mutilation.

To the elitists, I only have the following to say: Charles Chapman Pugh's Real Mathematical AnalysisJoseph Rotman's An Introduction to Algebraic Topology , anything by John Milnor, J.P.Serre or Jurgen Jost, Loring Tu's An Introduction to Manifolds, John McCleary's A First Course in Topology: Continuity and Dimension George F.Simmons' Differential Equations with Applications and Historical Notes,Charles Curtis' Linear Algebra: An Introductory Approach and John and Barbara Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach

I challenge them to consider any of these wonderful books to be spoon feeding students-and yet, they are eminently readable and wonderfully written books. In short, they are books students enjoy reading and therefore will not only learn from them-but will want to learn from them.



But an interesting trend has resulted from this myth. The students who are talented enough to learn from these texts who go on their careers to become mathematicians- and who care enough about teaching- recall their experiences as students. They don’t want to subject their students-or anyone's students-to the same torture. As a result, they try and write alternative books for students that do what they wish those texts had. The Computer Age has magnified this effect hundredfold as such books have become ridiculously easy to produce. As a result, we’ve gotten “backlash waves” of texts as alternatives to those classic tomes that created the large diversity of texts that currently exist in the various subfields of advanced mathematics. Where once there was a bare handful of such texts to choose from, a generation later, the “backlash” creates a myriad of them.

Some examples in the recent generations of math students will illustrate. Once, Alfhors’ ridiculously difficult Complex Analysis was the standard text in functions of a complex variable at U.S. graduate programs after the early 1960’s. There were a few alternatives available in English-such as Titchmarsh or Carathedory-but not a lot. This lead to an explosion of complex analysis texts in the 1970’s onward: Saks/Zygmund, Rudin, Bak/Newman, Conway, Heins, Greene/ Krantz, Jones/Singerman, Gamelin,- well, that list goes on and on. A similar backlash occurred in the 1960’s and 1970’s in general topology after an entire generation had suffered through John Kelley’s General Topology wrote a legion of such texts, including the classics by Willard and Munkres. This effect has further been enhanced by progress in those fields at the research level-which results in the presentations of the standard texts of a generation becoming outmoded. The result is the “backlash” presentations can also be “upgraded” to current language. A good example is the incorporation of category theory into advanced algebra texts post-1950’s.

I strongly believe the current large crop of graduate algebra texts is the result of a similar backlash against Lang.

I’ve gone on to some length about this because I think it’s important to keep these 2 ideas in mind- the elitist conception of Great Books and the backlash against it-when considering my readability criteria for judging such texts.

The list will be in 2 parts. The first part will focus on “warmup” texts i.e. texts that are generally too difficult to be considered first algebra texts for a standard undergraduate mathematics major, but too basic to be used as a text for a graduate course. I hope to write up a list of “basic” algebra texts for the usual students at some point. But for now, these are the books for the top students-those who have just finished honors calculus and are ready for a serious abstract mathematics course. The second part will be the heart of the list and will focus on first year graduate texts in strong programs.


So without further ado, my reading algebra list.

Enjoy.

And remember-comments and suggestions are not only welcomed, but encouraged.


Part I- Graduate Warmup: These are texts that are a little too difficult for the average undergraduate in mathematics, but aren’t quite comprehensive or rigorous enough for a strong graduate course. Of course, a lot of this is totally subjective. But it’ll make good suggestions for those struggling in graduate algebra because their backgrounds weren’t quite as strong as they thought.

Topics in Algebra, 2nd Edition by I.M.Herstein: This is the book I first learned algebra from under the sure hand of Kenneth Kramer at Queens College in his Math 337 course. It’s also the book that made me fall in love with the subject. Herstien’s style is concise yet awesomely clear at every step. His problem sets are legendarily difficult yet doable (mostly). If anyone asks me if they’re ready to take their algebra qualifier and how to prepare-I give them very simple advice: Get this book. If you can do 95 percent of the exercises, you’re ready for anything they throw at you. They’re THAT good. Warning: In true old European algebracist fashion, Herstein writes his functions in the very un-Calculus like manner on the right in composition i.e. fg= gof. This confused the author of this blog initially and no one corrected him until several weeks into the course-which lead to difficulties later on. A couple of quibbles with it-the field theory chapter is really lacking. The presentation, by today’s algebraists, may be considered somewhat old-fashioned. For example, Herstein doesn’t mention group actions and there are no commutative diagrams. This really hinders the presentation in some places. Also. Herstein tends to present even the examples-which are considerable- in their fullest generality. This makes the book harder for the beginner then it really needs to be. For example, he gives the dihedral group of rigid motions in the plane for the general n gon where n is an integer. he could start with the n=4 case and write out the full 8 member group table for the motions of the quadrilateral and then generalize. Still-I fell in love with this book. Many teachers of strong algebra courses today prefer either the more geometric approach of Artin or the similar but more modern and comprehensive approach of Dummit and Foote. Still, the book will always have a special place in my heart and I recommend it wholeheartedly for the talented beginner.

Algebra by Micheal Artin : The second edition of this book finally came out in Fall of 2010. For awhile, it looked like it might emerge posthumously-it was so long in gestation. But fortunately, this wasn’t the case. I must say in this revised review, the second edition is vastly improved over the first. The lack of exercises in the first edition has been greatly repaired with a host of new problems of varying levels of difficulty. He’s also reorganized and rewritten the book in many subtle ways that makes the writing and proofs much clearer then the first edition. Overall, the best qualities of the first edition have been preserved and improved upon. Its primary positive qualities are the heavily geometric bent and high level of presentation. The shift in emphasis from the permutation groups to matrix groups is an extremely smart one by Artin since it gives one a tool of much greater generality and simplicity while still preserving all the important properties of finite groups. (Indeed, permutations are usually explicitly represented as 2 x n matrices with integer valued bases-so the result is just a slight generalization. ) This also allows Artin to unify many different applications of algebraic structures to many different areas of mathematics-from classical geometry to Lie groups to basic topology and even some algebraic geometry (!) The major addition to the book’s presentation are many commutative diagrams allowing him to state most of the material in a completely modern manner. All through it, Artin brings an infectious love for algebra that comes through very sharply in his writing. Unfortunately, a lot of the flaws from the first edition still remain. Firstly, Artin assumes an awful lot of background in his prospective students-primarily linear algebra and basic Euclidean geometry. It might have been reasonable to assume this much background in the superhuman undergraduates at MIT in the early 1990’s, but I think that’s a stretch for most other students-even honors students. Especially nowadays. Secondly, the book is organized in a very idiosyncratic fashion that doesn’t always make sense even to people who know algebra. Nearly half the book is spent on linear algebra and group theory- rings, modules, fields are developed in a very rushed fashion. While Artin successfully expands a lot of these sections somewhat for the second edition, the book is still too unbalanced. And the discussion of modules is still too curt. Reading the second edition carefully also made me realize one of the major flaws of the overall style of the book-Artin can be painfully informal sometimes. For example, his discussion of cosets is actually confusing because he’s not as formal as he should be. It reminded me in a lot of ways of Allen Hatcher’s algebraic topology book, which suffers from a lot of the same informality. Lastly-his choice of topics for even good undergraduates is bizarre sometimes. He writes a chapter on group representations, but leaves tensor algebra and dual spaces “on the cutting room floor”? It’s a very strange choice. That being said, for all its flaws, a text of this level of daring and geometric focus by an expert of Artin’s stature is not to be ignored. I wouldn’t use it by itself, but I’d definitely keep a copy on my desk or on reserve for my students to browse.

A Course In Algebra by E.B. Vinberg This is very rapidly becoming my favorite reference for algebra. Translated from the Russian by Alexander Retakh, this book by one of the world’s preeminent algebracists is one of the best written, most comprehensive sources for undergraduate/graduate algebra that currently exists. Vinberg, like Artin, takes a very geometric approach to algebra and emphasizes the connections between it and other areas of mathematics. But Vinberg‘s book begins at a much more elementary level and gradually builds to a very high level indeed. It also eventually considers many topics not covered in Artin-including applications to physics such as the crystallographic groups and the role of Lie groups in differential geometry and mechanics! The most amazing thing about this book is how it manages to teach students such an enormous amount of algebra-from basic polynomial and linear algebra through Galois theory, multilinear algebra and concluding with the elements of representation theory and Lie groups, with an enormous number of examples and exercises that cannot be readily found in most other sources. All of it is done incredibly gently despite the steadily increasing sophistication of the material. The book has a very “Russian” style-by which I mean the author does not hesitate to both prove theorems and give applications to both geometry and physics (!) throughout. Those who know me personally know this is a position I am very sympathetic to-and for there to be a major recent abstract algebra text that takes this tack is very exciting to me.For anyone interested in writing a textbook on advanced mathematics, this is a terrific book to study for style. It is one of the most readable texts I have ever read. An absolutely first rate work that needs to be owned by any student learning algebra and any professor considering teaching it.

Abstract Algebra, 3rd edition by David S. Dummit and Richard M. Foote: Ever seen a movie or read a book where based on your tastes, everything you think and what you see in it, you should love it-but just the opposite? You don’t like it one bit and you couldn’t explain on pain of death why? That's how I feel about this book, one of the most popular and commonly used books for algebra courses-both undergraduate and graduate. It’s really frustrating that I feel that way because the book is really daring in its comprehensiveness and is surprisingly readable in many sections. It also has good exercises and more nice examples for the serious student then any book I've seen since Vinberg. So given all that, you'd think I'd be in love with the book, right? So do I, but I'm not. So what’s my problem with it? Well, first of all, it’s way too expensive. You could get both Vinberg AND a used copy of Artin for the same price as this book. (Yes, the price has come down considerably since I wrote this original review-but the book’s still ridiculously expensive brand new.) Second of all-it’s pretty dry and matter-of-fact. It just doesn’t excite me about algebra. Everything’s presented nicely and clearly-but it comes off almost like a dictionary. Lastly-the level the book is pitched at. It has pretty comprehensive coverage of the standard topics: groups, rings, field, and modules. I'm frustrated with my disappointment with D&F because the book has lots to offer.The group and field theory chapters in particular are outstanding and-I think- the highlights of the book. The book is also completely modern in outlook, it presents many commutative and functional diagrams also with many geometric examples a student will find very clarifying. It also contains some topics that are better suited for graduate courses- homological algebra and group representations, for example. The big problem is the book tries to cover all these topics at the same level and breadth as the basic material. As a result, it doesn’t succeed in developing these more sophisticated topics in enough depth for a graduate course. It also ends up covering way too much for any one-year undergraduate course. It’s certainly more comprehensive and modern then its ancestor text Herstein. So as a result, it ends up stuck in a weird level between undergraduate and graduate courses. I think this is probably what annoys me the most about this book-it comes off as a modernized and expanded,but bloated and watered down version of Herstien. Only about half the exercises are anywhere near as interesting as the ones there. I’d chop off section V altogether and expand it into a follow-up graduate text a la Knapp. I think that would a long way in improving the later sections-and the resulting 2 volume text has the potential to become the hands-down choice for the top colleges for their algebra courses. As is,it’s probably the best one book reference for algebra that currently exists and it’s nice to have handy for looking stuff up that you’ve forgotten or getting ready for exams. But it’s a problematic book to use for a course and needs to be used selectively.

OK, that ends part I. We get to the meat of the list in part II. Until then, my friends, may the forces of evil become confused on their way to your house. Ciao.




Some of the other books discussed in this post you definitely should check out: 




Leave a Reply