22
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ Would you be satisfied, or at least placated, by uses of the contrapositive? (Not that I can think of any right now, so this is purely hypothetical.) $\endgroup$ – Yemon Choi May 27 '15 at 19:35
  • $\begingroup$ @YemonChoi, an application of the contrapositive would be of interest to me once it is not just hypothetical. $\endgroup$ – KConrad Sep 2 '15 at 19:00
14
$\begingroup$

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!

$\endgroup$
14
$\begingroup$

I would be surprised if someone had an example of a ring which was only proved to be commutative using these results. I don't think that was the point of these theorems. I think they should be examined in the context of the time when they were proved.

At that time one of the fundamental problems in non-commutative algebra was the Burnside problem and its variants like the Kurosh problem.

These questions involved to what extent periodicity or torsion controlled the structure of an algebra. The original Burnside question asked whether a finitely generated periodic group must be finite. Work of Burnside and Schur showed this was the case for linear groups.

The Kurosh problem asked if every finitely generated nil algebra was finite dimensional or equivalently nilpotent. The Golod-Shafarevich construction led to a solution to both the Kurosh and unrestricted Burnside problem in the sixties.

In that context Jacobson was studying a different periodicity condition on the elements of an algebra than nilpotence and managed to show it had a strong algebraic consequence: commutativity. This is surprising at first sight and the proof as I recall it uses a fair amount of Jacobson's structure theories for rings.

$\endgroup$
  • 3
    $\begingroup$ What puzzles me is how this topic after Jacobson and Herstein took on a life of its own, with commutativity theorems becoming an end-goal in itself when, as far as I can tell, such theorems are never applied to anything. That's why I wrote my question. $\endgroup$ – KConrad May 28 '15 at 1:37
  • 4
    $\begingroup$ I think this happens frequently in math that topics take on a life of their own and develop in ways that are far removed from their origins. I find Jacobson's original theorem elegant on terms of its statement and nice because its proof used almost all machinery known in non-commutative ring theory at the time. Also it for into the framework of Burnside problems on vogue at the time. Then the subject probably became a cottage industry. $\endgroup$ – Benjamin Steinberg May 28 '15 at 1:40
  • 2
    $\begingroup$ I would agree that they most likely have no applications outside of their niche. Jacobson answered a very natural question and then people just kept pushing it forward without motivation. $\endgroup$ – Benjamin Steinberg May 28 '15 at 1:59
  • 2
    $\begingroup$ @KConrad: To answer your original question above, yes the techniques Jacobson developed are extremely useful. To give one example, one key step in the Jacobson result is using subdirect products. And subdirect products pop up all over the place, with lots of interesting theorems. The Jacobson radical is extremely useful and important. etc... $\endgroup$ – Pace Nielsen Jun 23 '15 at 18:46
  • 2
    $\begingroup$ @KConrad: No, I haven't seen the theorems themselves ever used to prove other interesting theorems. (By the way, the theorems don't prove Wedderburn's little theorem. That theorem is needed as the first step in the proof of the commtuativity theorems.) $\endgroup$ – Pace Nielsen Jun 27 '15 at 0:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.