Posner's first theorem and related identities for semiprime rings

We generalize Posner's first theorem and related identities to arbitrary semiprime rings. For instance, Posner's first theorem for semiprime rings is proved as follows: Let $R$ be a semiprime ring with extended centroid $C$, and let $\delta,D\colon R\to R$ be derivations. Then $\delta D$ is also a derivation if and only if there exist orthogonal idempotents $e_1,e_2,e_3\in C$, $e_1+e_2+e_3=1$, and $\lambda\in C$ such that $e_1D=0$, $e_2\delta=0$ and $e_3\big(\delta-\lambda D\big)=0$, where $e_2R$ is $2$-torsion free and $2e_3R=0$.