Let R be a prime ring of characteristic different from 2, with right Utumi quotient ring U and extended centroid C, and let f(x(1), ... , x(n)) be a multilinear polynomial over C, not central valued on R. Suppose that d is a derivation of R and G is a generalized derivation of R such that C(r(1), ... , r(n)))d(f(r(1), ... ,r(n)) vertical bar d(f(r(1), ... , r(n)))C(f(r(1), ... , r(n))) = 0 for all r(1), ... , r(n) epsilon R. Then either d = 0 or G = 0, unless when d is an inner derivation of R, there exists lambda epsilon C such that G(x) =lambda x, for all x epsilon R and f( x(1), ... , x(n))(2) is central valued on R.

### A Quadratic Differential Identity with Generalized Derivations on Multilinear Polynomials in Prime Rings

#### Abstract

Let R be a prime ring of characteristic different from 2, with right Utumi quotient ring U and extended centroid C, and let f(x(1), ... , x(n)) be a multilinear polynomial over C, not central valued on R. Suppose that d is a derivation of R and G is a generalized derivation of R such that C(r(1), ... , r(n)))d(f(r(1), ... ,r(n)) vertical bar d(f(r(1), ... , r(n)))C(f(r(1), ... , r(n))) = 0 for all r(1), ... , r(n) epsilon R. Then either d = 0 or G = 0, unless when d is an inner derivation of R, there exists lambda epsilon C such that G(x) =lambda x, for all x epsilon R and f( x(1), ... , x(n))(2) is central valued on R.
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2014
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11570/2720169`
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