Jordan (α, β)-derivations on triangular algebras and related mappings

Let R be a 2-torsion free commutative ring with identity, A, B be unital algebras over R. and M be a unital (A, B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let T = [(A)(0) (M)(B)] be the triangular algebra consisting of A, B and M, and let d be an R-linear mapping from T into itself. Suppose that A and B have only trivial idempotents. Then the following statements are equivalent: (1) d is a Jordan (alpha, beta)-derivation on T; (2) d is a Jordan triple (alpha, beta)-derivation on T; (3) d is an (alpha, beta)-derivation on T. Furthermore, a generalized version of this result is also given. We characterize the actions of automorphisms and skew derivations on the triangular algebra T. The structure of continuous (alpha, beta)derivations of triangular Banach algebras and that of generalized Jordan (alpha, beta)-derivations of upper triangular matrix algebras are described. (C) 2010 Elsevier Inc. All rights reserved.