Derivation which acts as a homomorphism or as an anti-homomorphism in a prime ring

Abstract

Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mappingSuppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U . The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R). Mathematics Subject Classification: 16W25,16N60, 16U80

The k-Derivation Acting as a k-Endomorphism and as an Anti-k-Endomorphism on Semiprime Nobusawa Gamma Ring

“…; whereas, the behavior of d is somewhat restricted in case of prime rings in the way that if d is a derivation of a prime ring R acting as a homomorphism or an antihomomorphism on a nonzero right ideal U of R, Here, we extend the above mentioned results following [1,2,4,8,9] in classical ring theory to those in gamma ring theory with k-derivation acting as a k-endomorphism or an anti-k-endomorphism on semiprime N  -rings. Our objective is to prove that…”

Generalized Derivations Acting as Homomorphisms and Anti-Homomorphisms of Gamma Rings

“…Analogously Rehman [4] extended the results for generalized derivation acting on nonzero ideals in case of prime rings. Recently Ali and Kumar [5] established the above mentioned result for generalized (θ,ϕ)-derivations in prime rings. By the same motivation, we extend the results in [4] of classical ring theory to the Γ-ring theory in the case of generalized derivation acts as a homomorphism and an anti-homomorphism of prime Γ-rings.…”

Generalized Skew Derivations as Jordan Homomorphisms on Multilinear Polynomials

“…Following this line of investigation, in [17] we obtain the following result: Let R be a prime ring, L a non-central Lie ideal of R and F a non-zero generalized derivation of R. If F acts as a Jordan homomorphism on L, then either F (x) = x for all x ∈ R, or char(R) = 2, R satisfies the standard identity s 4 (x 1 , x 2 , x 3 , x 4 ), L is commutative and u 2 ∈ Z(R) for any u ∈ L. Generalized derivations and generalized (α, β)-derivations as homomorphisms, anti-homomorphisms or Lie homomorphisms in prime rings, as well as derivations as homomorphisms or anti-homomorphisms in σ-prime rings, have also been discussed in [2,3,4,5,30,32,36].…”