The kernel and range inclusions of integral derivations in semiprime rings.
Let \(R\) be a semiprime ring with extended centroid \(C\), symmetric Martindale quotient ring \(Q\), and essential ideal \(I\). For any \(C\)-integral derivation \(D\) of \(R\), by the theory of differential identities, there is an associated inner derivation \(ad(a)\) for \(a\in Q\). For any other derivation \(\delta\) of \(R\), the authors characterize the inclusion of the images of \(D\) and \(\delta\), and of their kernels. Specifically they prove that \(\{x\in I\mid D(x)=0\}\subseteq\{x\in I\mid\delta(x)=0\}\) if and only if \(\delta=c_nD^n+\cdots+c_1D+ad(b)\) for some \(n\geq 1\), \(c_i\in C\), \(b\in C[a]\), and that \(\delta(I)\subseteq D(I)\) if and only if the last relation above holds with \(\sum_{i\geq 1}(-1)^iD^i(c_i)=0\).
scientific article; zbMATH DE number 5376928
scientific article