Bull. Korean Math. Soc. 2013; 50(5): 1651-1657
Printed September 1, 2013
https://doi.org/10.4134/BKMS.2013.50.5.1651
Copyright © The Korean Mathematical Society.
Basudeb Dhara, Sukhendu Kar, and Sachhidananda Mondal
Paschim Medinipur, Jadavpur University, Jadavpur University
Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$, $m (\geq 1), n (\geq 1)$ two fixed integers and $a\in R$. (i) If $a((d(x)y+xd(y)+d(y)x+yd(x))^{n}-(xy+yx))^{m}=0$ for all $x,y\in I$, then either $a=0$ or $R$ is commutative; (ii) If char$(R)\neq 2$ and $a((d(x)y+xd(y)+d(y)x+yd(x))^{n}-(xy+yx))\in Z(R)$ for all $x,y\in I$, then either $a=0$ or $R$ is commutative.
Keywords: prime ring, derivation, extended centroid
MSC numbers: 16W25, 16R50, 16N60
2009; 46(3): 599-605
2011; 48(5): 917-922
2021; 58(3): 659-668
2018; 55(2): 573-586
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd