Let R be a prime ring of characteristic different from 2 and 3, Q (r) its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n a parts per thousand yen 1 a fixed positive integer. Let alpha be an automorphism of the ring R. An additive map D: R -> R is called an alpha-derivation (or a skew derivation) on R if D(xy) = D(x)y + alpha(x)D(y) for all x, y a R. An additive mapping F: R -> R is called a generalized alpha-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + alpha(x)D(y) for all x, y a R. We prove that, if F is a nonzero generalized skew derivation of R such that F(x)x[F(x), x] (n) = 0 for any x a L, then either there exists lambda a C such that F(x) = lambda x for all x a R, or R aS dagger M (2)(C) and there exist a a Q (r) and lambda a C such that F(x) = ax + xa + lambda x for any x a R.
Annihilating and power-commuting generalized skew derivations on lie ideals in prime rings
DE FILIPPIS, Vincenzo
2016-01-01
Abstract
Let R be a prime ring of characteristic different from 2 and 3, Q (r) its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n a parts per thousand yen 1 a fixed positive integer. Let alpha be an automorphism of the ring R. An additive map D: R -> R is called an alpha-derivation (or a skew derivation) on R if D(xy) = D(x)y + alpha(x)D(y) for all x, y a R. An additive mapping F: R -> R is called a generalized alpha-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + alpha(x)D(y) for all x, y a R. We prove that, if F is a nonzero generalized skew derivation of R such that F(x)x[F(x), x] (n) = 0 for any x a L, then either there exists lambda a C such that F(x) = lambda x for all x a R, or R aS dagger M (2)(C) and there exist a a Q (r) and lambda a C such that F(x) = ax + xa + lambda x for any x a R.Pubblicazioni consigliate
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