The kernel and range inclusions of integral derivations in semiprime rings

Let R be a semiprime ring with extended centroid C and with symmetric Martindale quotient ring Q. For a derivation δ of R and an ideal I of R, define and . Let δ,δ′ be derivations of R such that δ is C-integral and has the associated X-inner derivation ad(a), where a Q. The main results of this paper (Theorem 3.5) are the following two equivalences: (1) KerI(δ) KerR(δ′) for an essential ideal I of R if and only if δ′=∑i 1μiδi+ad(b) for some μi C and some b C[a]. (2) for an essential ideal I of R if and only if δ′=∑i 1μiδi+ad(b) for some μi C with ∑i 1(−1)iδi(μi)=0 and some b C[a].