Example of a commutative ring where commutativity is not trivial

Question

In the book "Noncommutative Rings" from Herstein, there are some theorems which states propertied wich implies the commutativity of a ring. For example, one theorem states the following: If $R$ is a ring, in which for every $x,y \in R$ exists a natural number $n$ which is bigger than $1$, such that $(xy-yx)^{n} = (xy - yx)$. My question is, where can you use such theorems? Because in all commutative rings I can think of, commutativity is more or less elementary.

Answer 1

This is a part of very broad and deep theory of varieties of rings and PI-rings. Such statements are used, for example to prove that a variety is finitely based. See, for example, the paper "On some commutativity theorems of Herstein" by Howard Bell. There are $>16$ papers referring to that paper and to results of Herstein and somewhat earlier similar results of Jacobson.

Answer 2

Here's a concrete, albeit artificial example: take a finite field of order $p^n$ and find a representation of the field in terms of $n\times n$ matrices over the field of $p$ elements.

If you were just given a file with those matrices in it and were asked if they form a commutative ring under matrix multiplication, it would probably be far from obvious all those matrices commute with each other. But one could confirm that $x^{p^n}=x$ for every $x$ and conclude that the ring is commutative (by a commutativity theorem different than the one you mentioned.)

Computationally, checking commutativity by brute force requires $2p^{2n}$ matrix multiplications, but I have a suspicion that by using exponentiation tricks and cacheing products as they are observed would make the second method computationally less expensive.