Abstract
Let R be a ring and U be a Lie ideal of R. Suppose that σ, τ are endomorphisms of R. A family D = {dn}n ∈ Nof additive mappings dn:R → R is said to be a (σ,τ)- higher derivation of U into R if d0 = IR, the identity map on R and holds for all a, b ∈ U and for each n ∈ N. A family F = {fn}n ∈ Nof additive mappings fn:R → R is said to be a generalized (σ,τ)- higher derivation (resp. generalized Jordan (σ,τ)-higher derivation) of U into R if there exists a (σ,τ)- higher derivation D = {dn}n ∈ Nof U into R such that, f0 = IR and (resp. holds for all a, b ∈ U and for each n ∈ N. It can be easily observed that every generalized (σ,τ)-higher derivation of U into R is a generalized Jordan (σ,τ)-higher derivation of U into R but not conversely. In the present paper we shall obtain the conditions under which every generalized Jordan (σ,τ)- higher derivation of U into R is a generalized (σ,τ)-higher derivation of U into R.
Highlights
Throughout the present paper R will denote an associative ring with center Z(R)
In view of the above definition we introduce the analogue of (σ, τ )-higher derivation in a more general setting
The main objective of the present paper is to find the conditions on R under which every generalized Jordan (σ, τ )-higher derivation of R is a generalized (σ, τ )-higher derivation of R
Summary
Throughout the present paper R will denote an associative ring with center Z(R). For any x, y ∈ R denote the commutator xy−yx by [ x, y]. With this point of view an additive mapping F : R → R is said to be a generalized (σ , τ )-derivation on R if there exists a (σ , τ )-derivation d : R → R such that F(xy) = σ (x)d(y) + F(x)τ (y) holds for all x, y ∈ R For such an example let S be any ring and R =. A family F = {fn}n∈N of additive maps fn : R → R is said to be generalized (σ , τ )-higher derivation Let R be an algebra over the field of rationals and σ , τ be the endomorphisms of
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