A condition that implies commutativity

Question

Let $R$ be a ring. A notable theorem of N. Jacobson states that if the identity $x^{n}=x$ holds for every $x \in R$ and a fixed $n \geq 2$ then $R$ is a commutative ring.

The proof of the result for the cases $n=2, 3,4$ is the subject matter of several well-known exercises in Herstein's Topics in Algebra. The corresponding proofs rely heavily on "elementary" manipulations. For instance, the proof of the case $n=3$ can be done as follows:

1) If $a, b \in R$ are such that $ab= 0$ then $ba=0$.

2) $a^{2}$ and $-a^{2}$ belong to $\mathbf{Z}(R)$ for every $a \in R$.

3) Since $(a^{2}+a)^{3} = (a^{2}+a)^{2}+(a^{2}+a)^{2}$ it follows that

$a=a+a^{2}-a^{2} = (a+a^{2})^{3}-a^{2} = (a^{2}+a)^{2}+(a^{2}+a)^{2}-a^{2}$

and whence the result. ▮

Certainly, the mind can't but boggle at the succinctness of the above solution. Actually, it is the conciseness of this argument that has prompted me to pose the present question: is an analogous demonstration of the general theorem possible? The one that appears in [1] depends on some non-trivial structure theorems for division rings.

As usual, I thank you in advance for your insightful replies, reading suggestions, web links, etc...

References

[1] I. N. Herstein, Noncommutative rings, The Carus Mathematical Monographs, no. 15, Mathematical Association of America, 1968.

[2] I. N. Herstein, Álgebra Moderna, Ed. Trillas, págs. 112, 119, and 153.

Answer 1

For fixed $n \in \mathbb{N}$, Birkhoff's completeness theorem implies that such a proof must exist in the first-order equational theory of rings - as I mentioned here in a recent post. Many years ago Stan Burris told me that John Lawrence discovered such an equational proof that works uniformly for all $n$ (possibly also for Jacobson's form $x^{n(x)} = x$). I don't know if the proof is published yet, but some clues as to how it may proceed might be gleaned from their earlier joint [work [1]] 1

1 S. Burris and J. Lawrence, Term rewrite rules for finite fields.
International J. Algebra and Computation 1 (1991), 353-369. http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/fields3.pdf

Answer 2

Yes, it is possible to write down an equational proof for every $n$. This is covered in the preprint

Equational proofs of Jacobson's Theorem, arXiv:2310.05301 [math.RA]

The rough idea is to prove the reduction to prime characteristic $p$ and then to $n=p^k$ in a constructive way. Then, the (more or less equivalent) proofs by Dolan and Nagahara-Tominaga, which reduce the Theorem to Wedderburn's Theorem, can be made constructive as well. The paper discusses several examples. For example, every ring with $x^{2023}=x$ is commutative by an equational proof.

I have started this project in 2014, then continued in 2020, and now finished it (finally!) in 2023. Unfortunately, there are still some cases missing. Namely, if $\mathrm{gcd}(k,p^k-1) > 1$ (cf. MO/455808), the constructive Wedderburn Theorems $W_{p,k,T^m}$ from the paper (which are used to prove that $p^k$-rings of characteristic $p$ are commutative) can only be verified theoretically by a computer program.

The issue with using Birkhoff's Completeness Theorem (as mentioned in Bill's answer) is that it is a pure existence result (see MO/353564). It does not tell us how equational proofs look like.