Herstein's little book "Noncommutative Rings" has a chapter called Commutativity Theorems in which he proves results like Jacobson's theorem: if a ring (associative with identity, please) has the property that for each element $x$ there is an integer $n(x) > 1$ such that $x^{n(x)} = x$ then the ring is commutative. This is then generalized to use only the ring commutators $ab-ba$ in the role of $x$ or to fix the exponent and weaken $x^n = x$ to $x^n – x$ lying in the center of the ring. The conclusion is always that the ring is commutative.
My question, in brief, is: so what? Have these general commutativity theorems for rings ever had applications besides being a steppingstone in the proof of yet another commutativity theorem? Can such theorems be used to prove some rings are commutative that are not obviously commutative by just staring at them? I found the survey paper "Commutativity conditions for rings: 1950 — 2005" (see http://www.sciencedirect.com/science/article/pii/S072308690600034X) by Pinter-Lucke, but it indicates no real use for any of these theorems.
Don't tell me about finite division rings or Boolean rings being commutative, or applications of their commutativity. The proofs of their commutativity does not need the generality of theorems like Jacobson's.
Best Answer
The first profit I made of any kind off of mathematics was made off of a commutativity theorem.
In 1983, I was a first-year graduate student at Berkeley enrolled in T. Y. Lam's course Noncommutative Ring Theory. After he proved Jacobson's Theorem he reflected on how marvelous and simple the semantic proof was over a potential syntactic proof. He challenged us with the project of deducing commutativity from either $x^5=x$ or $x^6=x$ without appealing to the subdirect representation theorem. It was an informal challenge, not HW, but he added the incentive "If you do it, I'll give you a nickel!"
I did both cases, so he gave me a dime.
I still have it!