The extended centroid and \(X\)-inner automorphisms of Ore extensions (Q1183288)

Let \(R\) be a prime ring and let \(Q_ s(R)\) denotes its symmetric Martindale ring of quotients. This paper determines the center, extended centroid, and \(X\)-inner automorphisms of the Ore extension \(R[t;S,D]\). Furthermore, the following interesting result is obtained. Theorem. There exists a nonzero ideal \(I\) of \(R[t;S,D]\) with \(I\cap R=0\) if and only if there exist \(0\neq q_ n\), \(q_{n-1},\dots,q_ 0\), \(q\in Q_ s(R)\) and an endomorphism \(\theta\) of this quotient ring such that \(\sum^ n_{i=1} q_ iD^ i(x)+q(q_ 0 x-\theta(x)q_ 0)=0\) for all \(x\in Q_ s(R)\).