Let R be a prime ring with extended centroid C, G be 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 and k a fixed integer. If [G(f(x_i)),x_i]_k=0, for all x_i in I then: 1) either G(x)=ax for all x in R, with aI=cI for a suitable c in C; 2) or there exists an idempotent element e in Soc(RC) such that IC=eRC. In this case, a complete descritpion of polynomial identities satisfied by I is provided.
Generalized Derivations With Engel Condition On Multilinear Polynomials
DE FILIPPIS, Vincenzo
2009-01-01
Abstract
Let R be a prime ring with extended centroid C, G be 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 and k a fixed integer. If [G(f(x_i)),x_i]_k=0, for all x_i in I then: 1) either G(x)=ax for all x in R, with aI=cI for a suitable c in C; 2) or there exists an idempotent element e in Soc(RC) such that IC=eRC. In this case, a complete descritpion of polynomial identities satisfied by I is provided.File in questo prodotto:
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