The kernel and range inclusions of integral derivations in semiprime rings

摘要

Let R be a semiprime ring with extended centroid C and with symmetric Martindale quotient ring Q. For a derivation 3 of R and an ideal I of R, define Ker(I) (delta) =(def) {r is an element of I vertical bar delta(r) = 0} and R1(delta)=(def) delta(I). Let delta, delta' be derivations of R such that delta is C-integral and has the associated X-inner derivation ad(a), where a E Q. The main results of this paper (Theorem 3.5) are the following two equivalences: (1) Ker(I)(delta) subset of Ker(R)(delta') for an essential ideal I of R if and only if delta' = Sigma(i >= 1) mu(i)delta(I) + ad(b) for some mu(i) is an element of C and some b is an element of C vertical bar a vertical bar. (2)R-I(delta') subset of R-R(delta) for an essential ideal I of R if and only if delta' = Sigma(i >= 1) mu(i)delta(i)+ad(b) for some mu(i) is an element of C with Sigma(i >= 1)(-1)(i)delta(i)(mu(i))=0 and some b is an element of C vertical bar a vertical bar. (c) 2008 Elsevier Inc. All rights reserved.