Derivations modulo elementary operators

Let R be a prime ring with extended centroid C and symmetric Martindale quotient ring Qs(R). Suppose that Qs(R) contains a nontrivial idempotent e such that eR+Re⊆R. Let ϕ:R×R→RC+C be the bi-additive map (x,y)↦G(x)y+xH(y)+∑iaixbiyci, where G,H:R→R are additive maps and where ai,bi,ci∈RC+C are fixed. Suppose that ϕ is zero-product preserving, that is, ϕ(x,y)=0 for x,y∈R with xy=0. Then there exists a derivation δ:R→Qs(RC) such that both G and H are equal to δ plus elementary operators. Moreover, there is an additive map F:R→Qs(RC) such that ϕ(x,y)=F(xy) for all x,y∈R. The result is a natural generalization of several related theorems in the literature. Actually we prove some more general theorems.