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A NOTE ON GENERALIZED LEFT (theta, phi)-DERIVATIONS IN PRIME RINGS
Abstract
In this paper we describe generalized left (theta, phi)-derivations in prime rings, and prove that an additive mapping in a ring R acting as a homomorphism or anti-homomorphism on an additive subgroup S of R must be either a mapping acting as a homomorphism on S or a mapping acting as an anti-homomorphism on S, through which some related results are improved.
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Let R be a left faithful ringU its right Utumi quotient ring and ρ a dense right ideal of R. An additive map g: ρ → U is called a generalized derivation if there exists a derivation δ of ρ into U such that for all x,y∈ρ. In this note, we prove that there exists an element a∈ U such that for all x ∈ ρ. From this characterization, it is proved that if R is a semiprime ring and if g is a generalized derivation with nilpotent values of bounded index, then g = 0. Analogous results are also obtained for the case of generalized derivations with nilpotent values on Lie ideals or one-sided ideals.
Let R be a prime ring with characteristic different from two and S a non-empty subset of R. Suppose that θ, Φ are endomorphisms of R. An additive mapping F : R → R is called a generalized (θ, Φ)-derivation (resp. generalized Jordan (θ, Φ)-derivation) on S if there exists a (θ, Φ)-derivation d : R → R such that F(xy) = F(x)θ(y) + Φ(x)d(y) (resp. F(x ) = F(x)θ(x) + Φ(x)d(x)), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u ∈ U, for all u ∈ U. In the present paper, it is shown if θ is an automorphism of R then every generalized Jordan (θ, Φ)-derivation F on U is a generalized (θ, Φ)-derivation on U.
Let R be a ring and X be a left R-module. The purpose of this paper is to investigate additive mappings D1: R → X and D2: R → X that satisfy D1(ab) = aD1(b) + bD1(a), a, b ∈ R (left derivation) and D2(a2) = 2aD2(a), a ∈ R (Jordan left derivation). We show, by the rather weak assumptions, that the existence of a nonzero Jordan left derivation of R into X implies R is commutative. This result is used to prove two noncommutative extensions of the classical Singer-Wermer theorem.
