A note on skew derivations in prime rings
Bull. Korean Math. Soc. 2012 Vol. 49, No. 4, 885-898
https://doi.org/10.4134/BKMS.2012.49.4.885
Published online July 1, 2012
Vincenzo De Filippis and Ajda Fo\v sner
Faculty of Engineering University of Messina, University of Primorska
Abstract : Let $m, n, r$ be nonzero fixed positive integers, $R$ a $2$-torsion free prime ring, $Q$ its right Martindale quotient ring, and $L$ a non-central Lie ideal of $R$. Let $D:R\longrightarrow R$ be a skew derivation of $R$ and $E(x)=D(x^{m+n+r})-D(x^m)x^{n+r}-x^mD(x^n)x^{r}-x^{m+n}D(x^{r})$. We prove that if $E(x)=0$ for all $x\in L$, then $D$ is a usual derivation of $R$ or $R$ satisfies $s_4(x_1,\ldots,x_4)$, the standard identity of degree $4$.
Keywords : skew derivation, automorphism, prime ring
MSC numbers : 16W25, 16W20, 16N60
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