On \(\sigma\)-Twisted Functional Identities and Elementary Operators Over Division Rings
Main Article Content
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.
Article Details
References
- T.-K. Lee, J.-H. Lin, Certain Functional Identities on Division Rings, J. Algebra 647 (2024), 492–514. https://doi.org/10.1016/j.jalgebra.2024.03.002.
- W.S. Martindale III, Prime Rings Satisfying a Generalized Polynomial Identity, J. Algebra 12 (1969), 576–584. https://doi.org/10.1016/0021-8693(69)90029-5.
- M. Brešar, Introduction to Noncommutative Algebra, Springer International Publishing, 2014. https://doi.org/10.1007/978-3-319-08693-4.
- M. Brešar, M.A. Chebotar, W.S. Martindale III, Functional Identities, Birkhäuser Basel, 2007. https://doi.org/10.1007/978-3-7643-7796-0.
- L. Catalano, T. Merchán, On Rational Functional Identities, Commun. Algebra 52 (2024), 717–722. https://doi.org/10.1080/00927872.2023.2247488.
- T.Y. Lam, A First Course in Noncommutative Rings, Springer New York, 2001. https://doi.org/10.1007/978-1-4419-8616-0.
- N. Jacobson, Structure of Rings, revised ed., American Mathematical Society, 1964.