A zero-dimensional ring that is not von Neumann regular

An associative ring $R$ is called von Neumann regular if for each $x\in R$ there exists a $y\in R$ such that $x = xyx$.

Now let $R$ be a commutative ring. Its dimension is the supremum over lengths of chains of prime ideals in $R$. So for example, fields are zero dimensional because the only prime ideal in a field is the zero ideal.

Theorem. Let $R$ be a commutative ring. If $R$ is von Neumann regular, then it is zero dimensional.

The proof follows directly from the definition: suppose $P\subset R$ is a prime ideal of a von Neumann regular ring. If $x\not\in P$ and $y\in R$ is an element such that $x = xyx$, then $x(1 – yx) = 0$. Since $x\not\in P$, we must have $1 = yx$. Therefore, $P$ is maximal.

What about the converse? That’s what this counterexample is all about.
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A finitely generated flat module that is not projective

Let’s see an example of a finitely-generated flat module that is not projective!

What does this provide a counterexample to?

If $R$ is a ring that is either right Noetherian or a local ring (that is, has a unique maximal right ideal or equivalently, a unique maximal left ideal), then every finitely-generated flat right $R$-module is projective.

So what happens if we drop the Noetherian and local hypotheses?

The Example

Let $R = \prod_{j=1}^\infty F_j$ be an infinite product of fields and let $I = \oplus_{i=1}^\infty F_j$ be the ideal that is the direct sum of all the fields. Then the module $R/I$ is finitely generated. It is also flat, because $R$ is von Neumann regular and in such rings, every module is flat. Why is it not projective?

To see that it is not projective, consider the exact sequence
$$0\to I\to R\to R/I\to 0.$$ If $R/I$ were projective, that would mean that the map $R\to R/I$ splits, which gives a direct sum decomposition $I\oplus R/I\xrightarrow{\sim} R$ where the composition of the map $I\to I\oplus R/I\to R$ is the inclusion $I\to R$. The image of $R/I$ then corresponds to a nonzero ideal in $R$. But any nonzero ideal intersects $I$, so such a splitting is impossible.

This example is part of my new counterexamples project.

Kourkovka Notebook: Open problems in group theory

Every once in a while I spot a true gem on the arXiv. Unsolved Problems in Group Theory: The Kourkovka Notebook is such a gem: it is a huge collection of open problems in group theory. Started in 1965, this 19th volume contains hundreds of problems posed by mathematicians around the world. Additionally, problems solved from past volumes are also included with references.

For example, F.M. Markel proved that if $G$ is a finite supersolvable group with no two conjugacy classes having the same number of elements, then $G$ is actually isomorphic to the symmetric group $S_3$. Pretty cool right? Jiping Zhang extended this theorem by replacing ‘supersolvable’ by just ‘solvable’. Problem 16.3 in the Kourkovka notebook asks the obvious: if $G$ is any finite group where no two conjugacy classes have the same size, is $G\cong S_3$? There are of course many more problems of varying technicality, but there should be something in here for any group theorist.

I’ve always thought that you can gauge the health of a discipline by the quality of open problems in it. If that’s true, then the Kourkovka notebook shows that group theory is thriving very well.

Britton’s lemma and a non-Hopfian fp group

In a recent post on residually finite groups, I talked a bit about Hopfian groups. A group $G$ is Hopfian if every surjective group homomorphism $G\to G$ is an isomorphism. This concept connected back to residually finite groups because if a group $G$ is residually finite and finitely generated, then it is Hopfian. A free group on infinitely many generators is an example of a residually finite group that is not Hopfian.

Are there examples of finitely generated groups that are not Hopfian? Such an example would then of course give us an example of a group that is not residually finite.

In this post, we’ll see an example of a group that is finitely presented and not Hopfian. Not only that, but I promise the construction is actually not even scary, unlike those finitely presented groups with unsolvable word problem.
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How to make your own WordPress theme

This is a meta post on blogging, not mathematics. Recently, I got it into my head that I should design my own WordPress theme from scratch. As a consequence, you may have noticed that the theme of this blog has changed a little. I don’t know if many other math bloggers will want to try this (given that there aren’t too many active math bloggers these days), but if you really want the ultimate in customization, it’s the only way to go.

The reason why I did this is that I wasn’t totally happy with how customizable my last theme was, even though I like that theme generally. By designing your own WordPress theme, you can pretty much change anything and the final product is only limited by your own technical limitations. If you do design your own theme, you’ll realize that designing a WordPress theme is really not that scary after all. Given that, you need a little basic knowledge to get your theme started:
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Book Review: Riot at the Calc Exam by Colin Adams

When it comes to math humour, there’s not much out there. There is a good list of jokes on MathOverflow. There’s also Mathematical Apocrypha by Krantz, many of whose folklore stories are also amusing. The other day at the library I found another one: Riot at the Calc Exam by Colin Adams.
book cover
Adams’ book is a collection of 33 short stories involving teaching, exams, research, and the math anxiety of students. Most, but not all of these are strictly in the short story format. “The Theorem Blaster” for example is an advertisement for a machine to simplify theorems. One of my favourites, “The Mathematical Ethicist”, is a series of Dear X style letters with responses from an unethical ethicist.

So, is it funny? I found it to be so, and laughed aloud or smiled frequently. Most of the humour does indeed revolve around math themes, some of them advanced, so you really need an advanced math degree to get all the jokes on the first reading. That being said, there are end of chapter notes that I skimmed that give an explanation of some of the more esoteric concepts.

I liked also that the stories can be enjoyed on multiple levels. The “Deprogrammer’s Tale” is a story about young students being drawn into mathematics as though it were a vice to beware. This is one of the few stories that is less humour and more caricature of truth. Being currently looking for jobs myself, I have had recent experience with the odd view that non-academic employers take towards a math PhD that vaguely mirrors some of the reactions of the characters in this story.

Riot at the Calc Exam is certainly a fun read and I recommend it to anyone looking for a collection of good stories imbued with the curious theme of mathematics.

Commutators and the Ore Conjecture

In a talk yesterday by Boris Kunyavski at the University of Ottawa, I learned a little about the Ore conjecture, which in 2010 was proved a theorem in:

Liebeck, Martin W.; O’Brien, E. A.; Shalev, Aner; Tiep, Pham Huu. The Ore conjecture. J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939-1008.

It’s quite a fascinating result that arises by considering commutators in groups. If $G$ is a group, its commutator subgroup $[G,G]$ is the subgroup of $G$ generated by all the commutators $[g,h] = ghg^{-1}h^{-1}$ of $G$. It’s easy to see that the commutator subgroup is normal. A group $G$ is said to be perfect if $G = [G,G]$.

So let’s assume $G$ is perfect. This implies that every element of $G$ can be written as a product of commutators. But can every element of $G$ be written as a single commutator? That’s really far from obvious. For example, take your favourite perfect group and an element in it: can you prove that this single element is a commutator? Not so easy, right?

In fact, we can define the commutator length of any $g\in G$ to be the minimum number of commutators in all products of commutators equal to $g$. If $g$ can’t be written as the product of commutators, then its commutator length is infinite.

The commutator width of a group is defined to be the supremum over commutator lengths of all the elements of $G$. (Note: I think this should just be called the commutator length of $G$ as well, but that’s how the terminology ended up!)

It turns out that finding a perfect group $G$ with commutator width greater than one is quite tricky. In fact, the theorem proved in loc. cit. is:

Former Ore Conjecture/Now Theorem. If $G$ is a finite nonabelian simple group, then every element of $G$ is a commutator.

That’s pretty cool, though the proof is very long. That’s not surprising since it is a theorem about all finite nonabelian simple groups. What’s perhaps even more surprising is that there are examples of finitely-generated infinite simple groups containing elements that are not commutators. In fact, examples exist of such $G$ with infinite commutator length, as given in Alexey Muranov’s paper:

Finitely generated infinite simple groups of infinite square width and vanishing stable commutator length. J. Topol. Anal. 2 (2010), no. 3, 341-384.

This result makes use of small cancellation theory, which is a geometric group theory machinery that studies presented groups whose relations don’t have too much in common.

What is a residually finite group?

We say that a group $G$ is residually finite if for each $g\in G$ that is not equal to the identity of $G$, there exists a finite group $F$ and a group homomorphism
$$\varphi:G\to F$$ such that $\varphi(g)$ is not the identity of $F$.

The definition does not change if we require that $\varphi$ be surjective. Therefore, a group $G$ is residually finite if and only if for each $g\in G$ that is not the identity, there exists a finite index normal subgroup $N$ of $G$ such that $g\not\in N$.

Hence, if $G$ is residually finite, then the intersection of all finite-index normal subgroups is trivial. The converse holds, too (why?).
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Links to Atiyah’s preprints on the Riemann hypothesis

Sir Michael Atiyah’s preprints are now on the internet:

  1. The Riemann Hypothesis
  2. The Fine Structure Constant

The meat of the claimed proof of the Riemann hypothesis is in Atiyah’s construction of the Todd map $T:\C\to \C$. It supposedly comes from the composition of two different isomorphisms
$$\C\xrightarrow{t_+} C(A)\xrightarrow{t^{-1}_{-}} \C$$ of the complex field $\C$ with the $C(A)$, the center of a von Neumann hyperfinite factor $A$ of type II-1. Understanding Atiyah’s work boils down to understanding this Todd map, and therefore in understanding what is in the paper “The Fine Structure Constant”.

Assuming there is a zero $b$ of the Riemann zeta function $\zeta(s)$ off the critical line, Atiyah defines a function
$$F(s) = T(1 +\zeta(s + b)) – 1.$$ The function $F$ satisfies $F(0) = 0$ because one of the properties of the Todd map $T$ is that $T(1) = 1$, and $F$ is also supposedly analytic. According to some of the basic properties of the Todd function which is a polynomial on a closed rectangle containing this zero, this would imply that $F(s) = 0$ and therefore that $\zeta$ is identically zero, which is the contradiction.

Now, I have very little understanding of von Neumann algebras so I won’t comment at all on the Todd map. I have no doubt that the experts will dissect this because there’s so much attention on it. Even assuming all the properties of the Todd function, I find the proof difficult to follow. For example, the assumed zero $b$ off the critical strip: I can’t find where “$b$ is off the critical strip” is even being used. In fact, it’s hard to see where any of the basic properties of the zeta function are being used.