Generalized derivations on unital algebras determined by action on zero products

Let A be a unital algebra having a nontrivial idempotent and let M be a unitary A-bimodule. We consider linear maps F,G : A -> M satisfying F(x)y + xG(y) = 0 whenever x,y is an element of A are such that xy = 0. For example, when A is zero product determined algebra (e.g. algebra generated by idempotents) F and G are generalized derivations F(x) = F(1)x + D(x) and G(x) = xG(1) + D(x) for all x E A, where D : A -> M is a derivation. If A is not generated by idempotents then there exist also nonstandard solutions for maps F and G. In the case of A being a triangular algebra under some condition on bimodule M the characterization of maps F and G is given. We also consider conditions on algebra A making it a zero product determined algebra. (C) 2013 Elsevier Inc. All rights reserved.