The kernel and range inclusions of integral derivations in semiprime rings

Abstract

Let R be a semiprime ring with extended centroid C and with symmetric Martindale quotient ring Q . For a derivation δ of R andLet 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) Ker I (δ) ⊆ Ker R (δ ) for an essential ideal I of R if and only if δ = i 1 μ i δ i + ad(b) for some μ i ∈ C and some b ∈ C [a].(2) R I (δ ) ⊆ R R (δ) for an essential ideal I of R if and only if δ = i 1 μ i δ i + ad(b) for some μ i ∈ C with i 1 (−1) i δ i (μ i ) = 0 and some b ∈ C [a].