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)RI(δ′)⊆RR(δ) for an essential ideal I of R if and only if δ′=∑i⩾1μiδi+ad(b) for some μi∈C with ∑i⩾1i(−1)δi(μi)=0 and some b∈C[a].