Anti-automorphisms satisfying an Engel condition

ABSTRACT

Let R be a semiprime ring with an anti-automorphism τ, which is of finite order. It is proved that if $[[\dots \left[\tau \right(x),x^{n_1}],\dots ],x^{n_k}]=0$ for all x∈R, where $n_1,n_2,\dots ,n_k$ are k fixed positive integers, then τ is a commuting map. Moreover, commuting anti-automorphisms of semiprime rings are also characterized.

KEYWORDS:

• Anti-automorphism
• commuting map
• Engel condition
• involution
• m-power commuting map
• semiprime ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16N60
• 16R50

Acknowledgments

The author is grateful to the referee for the valuable suggestions which help to clarify and simplify the whole paper. In particular, the short proof of Lemma 2.8 is suggested by the referee.