Derivations with annihilator conditions in prime rings
Bull. Korean Math. Soc. 2013 Vol. 50, No. 5, 1651-1657
Printed September 1, 2013
Basudeb Dhara, Sukhendu Kar, and Sachhidananda Mondal
Paschim Medinipur, Jadavpur University, Jadavpur University
Abstract : 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
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