On Generalized Jordan Triple (σ, τ)-Higher Derivations in Prime Rings. (22nd January 2014) - Ashraf, Mohammad; Khan, Almas
Abstract : LetR be a ring and letU be a Lie ideal ofR . Suppose thatσ, τ are endomorphisms ofR, andℕ is the set of all nonnegative integers. A familyF = f n n ∈ ℕ of mappingsf n : R → R is said to be a generalized( σ, τ ) -higher derivation (resp., generalized Jordan triple( σ, τ ) -higher derivation) ofR if there exists a( σ, τ ) -higher derivationD = d n n ∈ ℕ ofR such thatf 0 = I R, the identity map onR, f n ( a + b ) = f n ( a ) + f n ( b ), andf n ( a b ) = ∑ i + j = n f i ( σ n - i ( a ) ) d j ( τ n - j ( b ) ) (resp., f n ( a b a ) = ∑ i + j + k = n f i ( σ n - i ( a ) ) d j ( σ k τ i ( b ) ) d k ( τ n - k ( a ) ) ) hold for alla, b ∈ R and for everyn ∈ ℕ . If the above conditions hold for alla, b ∈ U, thenF is said to be a generalized( σ, τ ) -higher derivation (resp., generalized Jordan triple( σ, τ ) -higher derivation) ofU intoR . In the present paper it is shown that ifU is a noncentral square closed Lie ideal of a prime ringR of characteristic different from two, then every generalized Jordan triple( σ, τ ) -higher derivation ofU intoR is a generalized( σ, τ ) -higher derivation ofU intoR .