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Power Values of Derivations on Multilinear Polynomials in Prime Rings
Basudeb Dhara
,
Sukhendu Kar
,
Sachhidananda Mondal
Abstract
Let $R$ be a prime ring with center $Z(R)$ and with extended centroid $C, d$ a derivation of $R$ and $f(x_1,\dots,x_n )$ a nonzero multilinear polynomial over $C$, $m \geq 1$ and $p \geq 1$ two integers. In the present paper, we study the situations (i) $((d(f(x_1,\dots,x_n )))^m − f(x_1,\dots,x_n ))^p = 0$; (ii) $((d(f(x_1,\dots,x_n )))^m − f(x_1,\dots,x_n ))^p \in Z(R)$ for all $x_1,\dots,x_n$ in some subsets of $R$.