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ANNIHILATORS OF POWER VALUES OF GENERALIZED DERIVATIONS ON MULTILINEAR POLYNOMIALS

Part of: Radicals and radical properties of rings Rings and algebras with additional structure

Published online by Cambridge University Press:  19 June 2009

VINCENZO DE FILIPPIS
VINCENZO DE FILIPPIS*
Affiliation:
Dipartimento di Scienze per l’Ingegneria e per l’Architettura, Faculty of Engineering, University of Messina, 98166, Messina, Italy (email: [email protected])
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Abstract

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Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f(x1,…,xn) be a noncentral multilinear polynomial over C, m≥1 a fixed integer, a a fixed element of R, g a generalized derivation of R. If ag(f(r1,…,rn))m=0 for all r1,…,rn∈I, then one of the following holds:

  1. (1) aI=ag(I)=(0);

  2. (2) g(x)=qx, for some q∈U and aqI=0;

  3. (3) [f(x1,…,xn),xn+1]xn+2 is an identity for I;

  4. (4) g(x)=cx+[q,x] for all x∈R, where c,q∈U such that cI=0 and [q,I]I=0.

Keywords

prime ringsdifferential identitiesgeneralized derivations

MSC classification

Secondary: 16N60: Prime and semiprime rings 16W25: Derivations, actions of Lie algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 2 , October 2009 , pp. 217 - 232
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

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