Generalized power commuting antiautomorphisms

Abstract

Let R be a noncommutative prime ring equipped with an antiautomorphism f such that (i) [f(x^k), x^m] = 0, all x $\in$ R, where k, m are fixed positive integers. In this article, we establish that from (i) follows (ii) [f(x), x = 0, all x $\in$ R. More is to be said about the type of f and the structure of R.

Keywords:

• Antiautomorphism
• commuting map
• generalized power commuting anti-automorphism prime ring scalar involution
• prime ring

2020 Mathematics Subject Classification:

• Primary: 16R20
• Secondary: 16W10

Acknowledgments

The author is indebted to T. K. Lee for bringing to his attention the result of C. L. Chuang quoted in Lemma 1.2. The author is also indebted to both referees 1 and 2 for their kind appreciations of this work and again to referee 2 for a very careful reading of the article.