\sigma-Lie ideals with derivations as homomorphisms and anti-homomorphisms

Abstract

Let R be a 2-torsion free σ-prime ring, U a nonzero σ-square closed Lie ideal of R and d a derivation of R which commutes with σ. If d acts as a homomorphism or an anti-homomorphism on U, then either d = 0 or U ⊆ Z(R).Mathematics Subject Classification: 16W10, 16W25, 16U80.

Jordan Derivations on Lie Ideals of ?-Prime Rings

“…On the other hand, various remarkable characterizations of -prime rings on -square closed Lie ideals have been studied by many authors viz. M. R. Khan, D. Arora and M. A. Khan [4] ; Oukhtite and Salhi [7,8,9,10] and J. Bergun, I. N. Herstein and J. W. Kerr [2] and I. N. Herstein [3]. In this paper, we shall prove that if d:R R is an additive mapping satisfying , where U is a -square closed Lie ideal of a 2-torsion free -prime ring R then for all and hence every Jordan derivations on a -prime ring R is a derivation on R. We begin with the following results.…”

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].…”

$\Gamma^*$-DERIVATION ACTING AS AN ENDOMORPHISM AND AS AN ANTI-ENDOMORPHISM IN SEMIPRIME $\Gamma$-RING M WITH INVOLUTION

“…al. in [6]. In Γ-rings, Dey and Paul [4] proved that if D is a generalized derivation of a prime Γ-ring M with an associated derivation d of M which acts as a homomorphism and an anti-homomorphism on a non-zero ideal I of M, then d = 0 or M is commutative.…”

“…In Γ-rings, Dey and Paul [4] proved that if D is a generalized derivation of a prime Γ-ring M with an associated derivation d of M which acts as a homomorphism and an anti-homomorphism on a non-zero ideal I of M, then d = 0 or M is commutative. Afterwards, Chakraborty and Paul [11] worked on kderivation of a semiprime Γ-ring in the sense of Nobusawa [10] and proved that d = 0 where d is a k-derivation acting as a k-endomorphism and as an anti-kendomorphism, the above mentioned results following [6] in classical rings are extended to those in gamma rings with derivation acting as a homomorphism and as an anti-homomorphism on σ-prime Γ-rings. In this paper we will prove that if d is Γ * -derivation of a semiprime Γ-ring with involution which is either an endomorphism or anti-endomorphism, then d=0.…”