Co-centralizing generalized derivations acting on multilinear polynomials in prime rings

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎Belda College‎, ‎Belda‎, ‎Paschim Medinipur‎, ‎721424‎, ‎W.B.‎, ‎India.

2 Department of Mathematics‎, ‎Jadavpur University‎, ‎Kolkata-700032‎, ‎W.B.‎, ‎India.

3 {Department of Mathematics‎, ‎Belda College‎, ‎Belda‎, ‎Paschim Medinipur‎, ‎721424‎, ‎W.B.‎, ‎India.

Abstract

‎Let $R$ be a noncommutative prime ring of‎ ‎characteristic different from $2$‎, ‎$U$ the Utumi quotient ring of $R$‎, ‎$C$ $(=Z(U))$ the extended centroid‎ ‎of $R$‎. ‎Let $0\neq a\in R$ and $f(x_1,\ldots,x_n)$ a multilinear‎ ‎polynomial over $C$ which is noncentral valued on $R$‎. ‎Suppose‎ ‎that $G$ and $H$ are two nonzero generalized derivations of $R$‎ ‎such that $a(H(f(x))f(x)-f(x)G(f(x)))\in C$ for all‎ ‎$x=(x_1,\ldots,x_n)\in R^n$‎. ‎ one of the following holds‎:
‎$f(x_1,\ldots,x_n)^2$ is central valued on $R$ and there exist $b,p,q\in U$ such‎ ‎that $H(x)=px+xb$ for all $x\in R$‎, ‎$G(x)=bx+xq$ for all $x\in R$ with $a(p-q)\in C$;‎
‎there exist $p,q\in U$ such that $H(x)=px+xq$ for all $x\in R$‎,
‎$G(x)=qx$ for all $x\in R$ with $ap=0$;‎
 $f(x_1,\ldots,x_n)^2$ is central valued on $R$ and there exist $q\in U$‎, ‎$\lambda\in C$ and an outer derivation $g$ of $U$‎
‎such that $H(x)=xq+\lambda x-g(x)$ for all $x\in R$‎, ‎$G(x)=qx+g(x)$ for all $x\in R$‎, ‎with $a\in C$;‎
$R$ satisfies $s_4$ and there exist $b,p\in U$ such‎ ‎that $H(x)=px+xb$ for all $x\in R$‎, ‎$G(x)=bx+xp$ for all $x\in R$‎.

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References

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Bulletin of the Iranian Mathematical Society
Volume 42, Issue 6
December 2016
Pages 1331-1342

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History

  • Receive Date: 04 April 2015
  • Revise Date: 25 August 2015
  • Accept Date: 26 August 2015

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