Let $K$ be a skew-field, infinite dimensional over its center $F$.

From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says that primitive PI-algebras have finite dimension over their center).

There, a GPI (generalized polynomial identity) has coefficients from the center $F$. If the coefficients are arbitrary, one has a GI (generalized identity), and a theorem by Amitsur describing the possible structure of $K$.

In my research, I now came upon skew-fields which might satisfy a "skew" GPI in the following sense: Let $\sigma$ be an involutory antiautomorphism of $K$ and $\gamma$ an automorphism of $K$ of order 1 or 2. For simplicity, let's restrict to the case that $\sigma$ and $\gamma$ commute.

Is it possible that $K$ satisfy an "skew" GPI, meaning that coefficients are arbitrary from $K$, and the GPI contains not just $x$, but also $x^\gamma$, $x^\sigma$ and $x^{\gamma\sigma}$ ? The case with a single unknown is all that interests me.

I actually kinda hope these skew fields don't exist respectively must be finite dimensional over their center.

  • $\begingroup$ The third sentence doesn't seem to parse (though it more or less clear what you want to say there). $\endgroup$ Aug 26 '11 at 13:35
  • $\begingroup$ There is array of results about identities of rings with involution (due to Amitsur and others). I am not absolutely sure whether also GPI with involutions of some sort were considered at certain point, but I would be surprised if they weren't (so your hope about finite-dimensionality - not sure what does "resp." means - seems to be plausible). $\endgroup$ Aug 26 '11 at 13:49
  • $\begingroup$ Fixed the third sentence, thanks (it's still a bit vague, but hopefully less confusing than before). $\endgroup$
    – Max Horn
    Aug 26 '11 at 17:24
  • $\begingroup$ "resp." meant "respectively", I clarified my question accordingly, thanks. As by your suggestion, I found some results by Amitsur regarding identities of rings with involutions and will take a look if they help (I wonder how I missed those before). Note that part of my problem is that there is both an involution $\sigma$ but also an automorphism $\gamma$ of order 2. But maybe I can reduce to a subfield where $\gamma$ is trivial (however, this subfield then better be stable under $\sigma$ and infinite) Thanks again! $\endgroup$
    – Max Horn
    Aug 26 '11 at 17:29

I recently came back to this question after leaving it aside for a few years, and this time my digging seems to have yielded an answer: Chen-Lian Chuang, "Differential identities with automorphisms and antiautomorphisms. I", Journal of Algebra 149, 371-404 (1992), proved the following theorem:

A prime ring satisfying a nontrivial differential identity with automorphisms and antiautomorphisms must also satisfy a nontrivial ordinary generalized polynomial identity (without derivations, automorphisms, and antiautomorphisms).

This applies in particular to division rings; in the language I used in the questions: A "skew GPI" implies the existence of a GPI.

As a corollary, any skew field admitting a non-trivial "skew GPI" must be finite-dimensional over its center.


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