Derivations modulo elementary operators

Let R be a prime ring with extended centroid C and symmetric Martindale quotient ring Q(s)(R). Suppose that Q(s)(R) contains a nontrivial idempotent e such that eR + Re subset of R. Let phi : R x R -> RC + C be the bi-additive map (x, y) -> G(x)y + xH(y) + Sigma(i)a(i)xb(i)yc(i), where G, H : R -> R are additive maps and where a(i), b(i), c(i) is an element of RC + C are fixed. Suppose that phi is zero-product preserving, that is, phi(x, y) = 0 for x, y is an element of R with xy = 0. Then there exists a derivation delta : R -> Q(s)(RC) such that both G and H are equal to delta plus elementary operators. Moreover, there is an additive map F : R -> Q(s)(RC) such that phi(x, y) = F(xy) for all x, y is an element of R. The result is a natural generalization of several related theorems in the literature. Actually we prove some more general theorems. (C) 2011 Elsevier Inc. All rights reserved.