Generalized derivations on unital algebras determined by action on zero products

Abstract 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 + x G ( y ) = 0 whenever x , y ∈ A are such that x y = 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 ) = x G ( 1 ) + D ( x ) for all x ∈ 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.