Let R be a prime ring with extended centroid C, g a generalized derivation of R, f(x_i) a multilinear polynomial over C in n non-commuting variables, I a non-zero right ideal of R. If [g(f(x_i)),x_i]=0, for all x_i in I, then 1) either g(x)=ax for all x in R, with aI=cI for a suitable element c in C; 2) or there exists an idmpotent element e in Soc(RC) such that IC=eRC. In this last case more informations about the polynomial identities satisfied by I are provided.
An Engel condition with generalized derivations on multilinear polynomials
DE FILIPPIS, Vincenzo
2007-01-01
Abstract
Let R be a prime ring with extended centroid C, g a generalized derivation of R, f(x_i) a multilinear polynomial over C in n non-commuting variables, I a non-zero right ideal of R. If [g(f(x_i)),x_i]=0, for all x_i in I, then 1) either g(x)=ax for all x in R, with aI=cI for a suitable element c in C; 2) or there exists an idmpotent element e in Soc(RC) such that IC=eRC. In this last case more informations about the polynomial identities satisfied by I are provided.File in questo prodotto:
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