On \(\sigma\)-Twisted Functional Identities and Elementary Operators Over Division Rings

Alsowait, Nawaf

doi:10.28924/2291-8639-24-2026-288

Research Article

On \(\sigma\)-Twisted Functional Identities and Elementary Operators Over Division Rings

Nawaf Alsowait

Abstract

Let \(\mathcal{D}\) be a division ring with centre \(\mathcal{Z}(\mathcal{D})\) and \(\sigma\) be an automorphism of \(\mathcal{D}\). For additive maps \(f:\mathcal{D}\to\mathcal{D}\), we study the functional identity \(\sigma(x)G(x)f(x)=H(x)\), where \(G(X)\) and \(H(X)\) are non-zero generalized polynomials in the variable \(X\) over \(\mathcal{D}\). By applying the inverse automorphism, we reduce the problem to the classical case and prove that either \([\mathcal{D}:\mathcal{Z}(\mathcal{D})]<\infty\) or \(f\) is a \(\sigma\)-elementary operator, that is, a finite sum of terms \(a\,\sigma(x)\,b\) with \(a,b\in\mathcal{D}\). We also consider additive solutions of \(f(x)=\sigma(x)^n g(x^{-1})\) for \(n>2\) and obtain the corresponding forms under the stated hypotheses.

How to Cite

N. Alsowait, On \(\sigma\)-Twisted Functional Identities and Elementary Operators Over Division Rings, Int. J. Anal. Appl. 24 (2026), 288. https://doi.org/10.28924/2291-8639-24-2026-288

BibTeX

@article{Alsowait2026,
  author  = {Alsowait, Nawaf},
  title   = {On \(\sigma\)-Twisted Functional Identities and Elementary Operators Over Division Rings},
  journal = {Int. J. Anal. Appl.},
  volume  = {24},
  year    = {2026},
  pages   = {288},
  doi     = {10.28924/2291-8639-24-2026-288}
}

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