Posner´s Second Theorem and an Annihilator Condition with Generalized Derivations

Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, d\neq 0 a non-zero generalized derivation of R, f(x1,..,xn) a non-central multilinear polynomial over C in n non-commuting variables, a \in R such that a[d(f(r1,..,rn)),f(r1,..,rn)]=0, for any r1,..,rn \in R. Then one of the following holds: 1. a=0; 2. there exists l \in C such that d(x)=l x, for all x \in R; 3. there exist q \in U and l \in C such that d(x)=(q+l)x+xq, for all x\in R, and f(x1,..,xn)2 is central valued on R.

Anahtar Kelimeler:

Prime rings, derivations, left Utumi quotient rings, two-sided Martindale quotient ring, differential identities

Posner´s Second Theorem and an Annihilator Condition with Generalized Derivations

Keywords:

Prime rings, derivations, left Utumi quotient rings, two-sided Martindale quotient ring, differential identities,

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Vincenzo DE FILIPPIS DI.S.I.A. Faculty of Engineering University of Messina Contrada Di Dio, 98166 Messina-ITALY e-mail:deŞlippis@unime.it