# Commutativity Theorems for Rings

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.

## 3 thoughts on “Commutativity Theorems for Rings”

1. That is very cool.

Any idea who the dude with the hat is, when you click on the first link?

2. 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.)

3. Thanks for the kind words. The guy with the hat is Emil Artin by the way.