# Generalized derivations acting on multilinear polynomials in prime rings

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## Abstract

Let for all

*R*be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring*U*and extended centroid*C*, let*F*,*G*and*H*be three generalized derivations of*R*,*I*an ideal of*R*and*f*(*x*_{1},...,*x*_{ n }) a multilinear polynomial over*C*which is not central valued on*R*. If$$F(f(r))G(f(r)) = H(f(r)^2 )$$

*r*= (*r*_{1},...,*r*_{ n }) ∈*I*^{ n }, then one of the following conditions holds:- (1)
there exist

*a*∈*C*and*b*∈*U*such that*F*(*x*) =*ax*,*G*(*x*) =*xb*and*H*(*x*) =*xab*for all*x*∈*R* - (2)
there exist

*a*,*b*∈*U*such that*F*(*x*) =*xa*,*G*(*x*) =*bx*and*H*(*x*) =*abx*for all*x*∈*R*, with*ab*∈*C* - (3)
there exist

*b*∈*C*and*a*∈*U*such that*F*(*x*) =*ax*,*G*(*x*) =*bx*and*H*(*x*) =*abx*for all*x*∈*R* - (4)
*f*(*x*_{1},...,*x*_{ n })^{2}is central valued on*R*and one of the following conditions holds- (a)
there exist

*a*,*b*,*p*,*p*’ ∈*U*such that*F*(*x*) =*ax*,*G*(*x*) =*xb*and*H*(*x*) =*px*+*xp*’ for all*x*∈*R*, with*ab*=*p*+*p*’ - (b)
there exist

*a*,*b*,*p*,*p*’ ∈*U*such that*F*(*x*) =*xa*,*G*(*x*) =*bx*and*H*(*x*) =*px*+*xp*’ for all*x*∈*R*, with*p*+*p*’ = ab ∈*C*.

- (a)

## MSC 2010

prime ring derivation generalized derivation extended centroid Utumi quotient ring## MSC 2010

16W25 16N60## Preview

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© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017