Higher derivatives and finiteness in rings (Q2711332)

Let \(R\) be a prime ring, \(I\neq 0\) an ideal of \(R\), \(T\neq 0\) a right ideal of \(R\), \(D\) a derivation of \(R\), and \(n>0\) a fixed positive integer. The main results of the paper show that when \(D^n(I)\) is finite, or when \(D^n(T)\) is finite and \(R\) satisfies a polynomial identity, then \(R\) is finite or \(D\) is nilpotent on \(R\). When \(R\) is semiprime without a nonzero finite right ideal, if \(D^n(T)\) is finite then \(D\) is periodic on a nonzero right ideal \(T_1\) of \(R\) contained in \(T\).