Let Rbe a ring with its center Z(R) and / a nonzero ideal of R. The purpose of this paper is to investigate identities satisfied by additive mappings on prime and semiprime rings. More precisely, we prove the following result. Let R be a semiprime ring, and let F, d : R → R be two additive mappings such that F(xy) = F(x)y + xd(y) for all x, y ∈ R. If F{xy) ±xy ∈ Z(R) for all x,y ∈ I, then [d{x),x] = 0 for all x ∈ /. Further, if d is a derivation such that d(I) ≠ (0), then R contains a nonzero central ideal. Moreover, if R is prime and d is a derivation such that d(I) ≠ (0), then R is commutative.
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