Anti-endomorphisms and endomorphisms satisfying an Engel condition

Abstract

Let R be a semiprime ring with τ an anti-endomorphism or endomorphism. It is proved that if τ satisfies an Engel condition $$\left[\right[\dots \left[\right[x^{\tau },x^{n_1}],x^{n_2}]\dots ],x^{n_k}]=0$$ for all $x\in R$, where $n_1,n_2,\dots ,n_k$ are k fixed positive integers, then τ is a commuting map (i.e. $[x^{\tau },x]=0$ for all $x\in R$). The theorem generalizes the results proved in [22] with τ an anti-automorphism of finite order and in [10, 11] with R a division ring, respectively.

Keywords:

• (Anti-)endomorphism
• Engel condition
• power commuting map
• semi-linear map
• semiprime ring
• prime ring

Mathematics Subject Classification 2010:

• 16R60
• 16N60

Acknowledgment

The authors are grateful to the referee for the valuable suggestions which help to clarify the article.

Additional information

Funding

The work of T.-K. Lee and J.-H. Lin was supported in part by the Ministry of Science and Technology of Taiwan (MOST 107-2115-M-002 -018 -MY2) and the National Center for Theoretical Sciences (NCTS), Taipei Office.