Nonlinear generalized Jordan (σ, Γ)-derivations on triangular algebras

• Main Authors: Alkenani Ahmad N., Ashraf Mohammad, Jabeen Aisha
• Format: Article
• Language: English
• Published: De Gruyter 2018-05-01
• Series: Special Matrices
• Subjects: triangular algebra; generalized jordan (σ; γ)-derivation; generalized (σ; 16w25; 15a78
• Online Access: https://doi.org/10.1515/spma-2017-0008

Let R be a commutative ring with identity element, A and B be unital algebras over R and let M be (A,B)-bimodule which is faithful as a left A-module and also faithful as a right B-module. Suppose that A = Tri(A,M,B) is a triangular algebra which is 2-torsion free and σ, Γ be automorphisms of A. A map δ:A→A (not necessarily linear) is called a multiplicative generalized (σ, Γ)-derivation (resp. multiplicative generalized Jordan (σ, Γ)-derivation) on A associated with a (σ, Γ)-derivation (resp. Jordan (σ, Γ)-derivation) d on A if δ(xy) = δ(x)r(y) + σ(x)d(y) (resp. σ(x²) = δ(x)r(x) + δ(x)d(x)) holds for all x, y Є A. In the present paper it is shown that if δ:A→A is a multiplicative generalized Jordan (σ, Γ)-derivation on A, then δ is an additive generalized (σ, Γ)-derivation on A.