N-commuting mappings on (semi)-prime rings with applications

Abstract

Let $\mathrm{ℛ}$ be a ring and ${\mathbb{Z}}^{+}$ be the set of positive integers. Suppose $\xi :\mathrm{ℛ}\to \mathrm{ℛ}$ is an automorphism of $\mathrm{ℛ}$. In this paper, we study the following functional identity $$x^{\xi }{x}^n+x^n{x}^{\xi }=0 \text{for} \text{every} x\in \mathrm{ℛ}, \text{and} \mathit{n}\in {\mathbb{Z}}^{+}.$$

As an application, we describe the structure of ${\mathrm{𝒞}}^{*}$-algebras.

Keywords:

• Prime and semiprime ring
• ${\mathrm{𝒞}}^{*}$-algebra
• n-skew commuting mapping
• automorphism
• maximal right ring of quotients
• generalized polynomial identity (GPI)

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W25
• 16N60
• 16R50
• 46L57

Acknowledgement

This Project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant No. $G-310-662-38$. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

Disclosure statement

No potential conflict of interest was reported by the authors.