Abstract
In this paper the centralizing and commuting concerning skew left -derivations and skew left -derivations associated with antiautomorphism on prime and semiprime rings were studied and the commutativity of Lie ideal under certain conditions were proved.
Highlights
Throughout this paper R represents an associative ring with center Z(R) and α∗ an antiautomorphism of R
Recall that a ring R is said to be prime if aRb=0 implies that either a=0 or b=0 for all a, b∈R (3) and it is semiprime if aRa=0 implies that a=0 for all a ∈ R (1)
An additive mapping ξ:R → R is called a derivation if ξ(υγ)= ξ(υ)γ + υ ξ(γ) for all υ, γ∈R (4), and it is called a skew derivation (α∗derivation) of R associated with the antiautomorphism α∗ if ξ(υγ)= ξ(υ)α∗(γ)+ υ ξ(γ) for all υ, γ∈R (5)
Summary
Throughout this paper R represents an associative ring with center Z(R) and α∗ an antiautomorphism of R. An additive mapping ξ:R → R is called a left derivation if ξ(υγ)= γξ(υ)+ υ ξ(γ) for all υ, γ∈R (6), and it is called a skew left derivation of R associated with antiautomorphism α∗ if ξ(υγ)=α∗(γ)ξ(υ)+ υ ξ(γ) for all υ, γ∈R (7), it is clear that the concepts of derivation and left derivation are identical whenever R is commutative. Definition (2.1) (2) A map ξ: Rn → R is called permuting (or symmetric) if the equation ξ(υ1, υ2, ...
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