Posted by Jason Polak on 28. January 2012 · Write a comment · Categories: books, modules · Tags: , ,

As it happens every so often, I browse the mathematical library pseudorandomly, and look out for interesting titles; usually a prerequisite for interesting is that they have something to do with the realm of algebra. This is exactly how I found Faith’s book, with its captivating title urging me to borrow it.

Now, inevitably in mathematical research, one has to efficiently skim through papers and books to find specific ideas and facts. The unfortunate thing is that sometimes it is easy to neglect the stimulation of the idle curiosity that probably brought most mathematicians into their fields in the first place, and so I try to combat this neglect by my idle browsing and blogging.

I try not to spend too much time on this so that I progress with my degree, but I try to nurture my curiosity through reading anything that looks interesting. Returning to books, I do believe there are few worse literary follies than a graduate algebra textbook that lacks imagination in its examples and theorems and passion in its explication. I only fear that such books will tend to promote in the learning of higher algebra what most institutions have done with calculus, and that is to make it a tiresome mechanical effort, washing away the once vibrant and fanciful colours from the gentle tendrils of the mind.

But fear not! Should the mental dessication start to occur in a young algebraist’s mind; should the flames of passion dim for the wonders of the injective module, she can always turn to the entire object of this post, videlicet Faith’s “Rings and Things and a Fine Array of Twentieth Century Associative Algebra”
. I refer to the second edition, incidentally, which corrects many errors from the 1st edition.

Let me say a word on authors. One of my undergraduate professors once told me that if at all possible, I should choose books in any specified topic by the depth of the author. Whatever this “depth” is, for an algebra book it certainly means that the author should have a deep knowledge of esoteric algebra and a passion for it. Undoubtedly, Carl Faith has both, and this can been seen not only from the review title but also from his two-volume algebra textbook, which I’ve had the joy to peruse on a few occasions.

At the risk of going on forever, and that risk is indeed high, let me start by describing this book itself. First, there are two parts of this book. The first half is general survey of the results and theorems of rings a modules, and the second is the author’s personal account of many of the mathematicians and fellow students he met throughout his career.

The First Part

The beauty of this book is that anyone with a basic understanding of ring theory and module theory will be able to appreciate virtually every theorem and result stated in this book. The format is by topic, and then in each topic a list of theorems are given, occasionally with proofs if the proof is short, and in any case references are given for all the theorems.

The theorems themselves are, as the book title suggests, results about rings and modules from the twentieth century. Furthermore, many of these theorems I believe will be interesting to students interested in algebra even if ring theory and module theory is not their primary passion. Of course, the reader should take into account that I have an unusual fascination with modules, perhaps because modules are everything vector spaces failed to be (that is, interesting).

Moreover, this results given in Rings and Things are not only interesting, but new to me, so that I am practically spellbound by seeing all these new theorems. The first theorem actually that I came across actually was the Perlis-Walker theorem:

Theorem. If $ G$ and $ H$ are finite abelian groups such that $ \mathbb{Q}G\cong \mathbb{Q}H$ then $ G\cong H$.

Here, $ \mathbb{Q}G$ denotes the group algebra. This is such a fascinating theorem that I wonder why it was never mentioned in my graduate algebra class when group rings were introduced. Of course, the answer is clear, in that at the time it was not used for the development of the material, and that I can understand. I only mention this because I believe the worth of mentioning these theorems is that they are the primary means to spark a fiery curiosity about the mathematical world.

I will just give two other examples of the results given in this book. Let us recall first that it is easy to see that $ \mathbb{Q}$ is not a free $ \mathbb{Z}$-module, in fact, it doesn’t take too long to prove that $ \mathbb{Q}$ is not projective either (but it is flat!). A more interesting result contained in Faith’s book, is Baer’s (which in turn is a special case of a corollary of Chase’s):

Theorem. $ \mathbb{Z}^\omega$ is not free.

However, a related theorem given below says:

Theorem. The Specker group of $ \mathbb{Z}^\alpha$ is free for any cardinal $ \alpha$.

The Specker group of $ \mathbb{Z}^\alpha$, as defined in Rings and Things, is the subgroup of all bounded sequences in $ \mathbb{Z}^\alpha$.

Here is a list of chapters of the first part:

  1. Direct Products and Sums of Rings and Modules and the Structure of Fields
  2. Introduction to Ring Theory: Schur’s Lemma and Semisimple Rings, Prime and Primitive Rings, Noetherian and Artinian Modules, Nil, Prime and Jacobson Radicals
  3. Direct Decompositions of Projective and Injective Modules
  4. Direct Product Decompositions of von Neumann Regular Rings and Self-injective Rings
  5. Direct Sums of Cyclic Modules
  6. When Injectives are Flat: Coherent FP-injective Rings
  7. Direct Decompositions and Dual Generalizations of Noetherian Rings
  8. Completely Decomposable Modules and the Krull-Schmidt-Azumaya Theorem
  9. Polynomial Rings over Vamosian and Kerr Rings, Valuation Rings and Prufer Rings
  10. Isomorphic Polynomial Rings and Matrix Rings
  11. Group Rings and Maschke’s Theorem Revisted
  12. Maximal Quotient Rings
  13. Morita Duality and Dual Rings
  14. Krull and Global Dimension
  15. Polynomial Identities and PI-rings
  16. Unions of Primes, Prime Avoidance, Associated Prime Ideals, Acc on Irreducible Ideals, and Annihilator Ideals in Commutative Rings
  17. Dedekind’s Theorem on the Independence of Automorphisms Revisted

The Second Part

This part of the book is Faith’s recollections of some of his interactions with various people, mostly mathematicians, most of which take place at Princeton and Rutgers. Amongst others, he recalls Armand Borel, Paul Cohen, Kurt Godel, Johnny von Neumann, Serge Lang (I found his interaction with Lang to be particularly amusing), and Irving Kaplansky (who is another of my favourite authors, and whose retiring presidential address for the AMS inspired the title of Faith’s book). There is very little actual mathematics in the second part, which I find appropriate since the focus of the second part seemed to be on human interaction.

I finished reading the second part in two days of some light nighttime reading and I was fascinated by the atmosphere surrounding Princeton during those times.

Final Remarks

As far as I know, Faith’s book is the only book that contains so much generally interesting recent algebra, not to mention in such an easily readable format. Combined with his personal accounts, this is definitely one book I must have on my shelf, and one book I think all interested in the intricacies of modules should spend significant amounts of time studying.

Edit: I communicated my review to Carl Faith and he was kind enough to respond with some comments, because of which I added the origins of the book title and a note about the second edition.

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