Generalized derivations on unital algebras determined by action on zero products

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