MathJax doesn’t work with RSS readers, so when I have some more time I will look into using a plug-in instead. Until then, sorry for filling up your RSS feed with dollar signs.
Theo Raedschelders has written a nice sketch of Herstein’s commutativity theorem for rings. It is a generalization of Wedderburn’s theorem that a finite division ring must be a field. The theorem states that if for every pair of elements $a$ and $b$ there exists an $n > 1$ (which can depend on $a$ and $b$) such that
$$
(ab – ba)^n = ab – ba,
$$
then the ring is commutative. What’s surprising about the proof is its indirectness. The proof requires essentially all of Nathan Jacobson’s structure theory for rings.
PlanetMath has a nice summary of known conditions that imply commutativity.
That is very cool.
Any idea who the dude with the hat is, when you click on the first link?
I don’t recognize him. The image name is “wheel.jpg”, which doesn’t tell me anything.
(If anyone else wants to take a look, it’s a rotating banner, so you don’t get the guy with the hat every time.)
Thanks for the kind words. The guy with the hat is Emil Artin by the way.