• HOMOMORPHISM OR ANTI- HOMOMORPHISM OF LEFT (α, 1) - DERIVATIONS IN PRIME RINGS

Abstract

Let R be a 2- torsion free ring and let U be a Lie ideal of R. Suppose that α, 1 are automorphisms of R. An additive mapping d: R→ R is said to be a left (α, 1)-derivation (resp. Jordan left (α, 1)-derivation) of R if d(xy) = α(x)d(y) + yd(x) (resp. d(x²) = α(x)d(x) + xd(x)) holds for all x, y∈ R. In this paper it is established that if R admits a nonzero left (α, 1)-derivation which acts as a homomorphism or as an anti-homomorphism on I of R, then       d = 0 on R. Also we prove that if G: R→ R is an additive mapping satisfying G(xy) = α(x)G(y) + yd(x) for all x, y∈ R and a left (α, 1)-derivation d of R such that G also acts as a homomorphism or as an anti-homomorphism on a nonzero ideal I of R, then either R is commutative or d = 0 on R.