On n^th power ∗-property in ∗-rings with applications

Abstract

Motivated by Dar and Ali’s [9] elegant work's, the goal of the paper is to study n^th power *-property in rings with involution and its related problems. In fact, the main result of this paper gives conditions which ensure that an additive map $d:\mathit{R}\to \mathit{R}$ satisfying n^th power *-property must be X-inner. As an application, we characterize additive map $F:\mathit{R}\to \mathit{R}$ such that $$F(x^n)=F(x){(x^{\ast })}^{n-1}+{\sum }_{i=1}^{n-1}{x}^{i}d(x){(x^{\ast })}^{n-i-1}\text{for}\text{all}\mathrm{x}\in \mathit{R}.$$ where $d:\mathit{R}\to \mathit{R}$ is an additive map satisfying n^th power *-property. These results provide the affirmative answer to the open problem (see Problem 3.3 [9]).

Keywords:

• Associative ring
• involution
• generalized Jordan *-derivation
• Jordan *-derivation
• Jordan left *-centralizer
• nth power *-property

2020 Mathematics Subject Classification:

• 16W10
• 16W25
• 16N60

Acknowledgements

The authors express their appreciation to the anonymous referee(s) whose feedback helped us to prepare a better final version of this paper. We would like to thank Professor Shakir Ali for his continuous encouragement, valuable suggestions for doing good research and pointing out our attention towards the open problems posed in his very recent paper [9]. We really appreciate his constant support and guidance during the preparation of this manuscript.