More on involutions with local Engel or power commuting conditions

ABSTRACT

Let R be a ring with * and with 1. Assume that R has no nil ideals (other than 0) and that R is integral over its center Z, that is to say, that each x in R satisfies a monic polynomial equation (in x) with coefficients in Z. Then the following conditions are equivalent. Condition 1: * is a commuting (or normal) involution, that is, for each x in R, $x^{\ast }x=x{x}^{\ast }$. Condition 2: For each x in R, there is an integer N = N(x)≥1 depending on x such that $d_x^N\left(x^{\ast }\right)=0$ where d_x is the map of R defined by d_x(y): = yx−xy, for all y in R, and $d_x^N$ is the N^th power of d_x (under composition). Condition 3: For each x in R, there is an integer $N^{\prime }=N^{\prime }\left(x\right)=1$ depending on x such that $x^{\ast }{x}^{N\prime }=x^{N\prime }{x}^{\ast }$.

KEYWORDS:

• Algebraic over the center
• commuting involution
• integral over the center
• local Engel condition
• local power commuting condition

AMS SUBJECT CLASSIFICATION:

• Primary 16R20
• Secondary 16W10/16R60

Acknowledgment

The author would like to thank the referee for careful reading of the paper.