Generalized derivations with nilpotent values on semiprime rings.
This paper concerns a generalized derivation \(g\colon I\to R\) for \(R\) a semiprime ring, \(I\) an essential ideal of \(R\), and \(d\colon I\to R\) the derivation associated to \(g\). The main results show that \(g\) extends to a generalized derivation of the left Martindale quotient ring \(R_F\) of \(R\) and has the form \(g(x)=ax+d(x)\) for some \(a\in R_F\), \(d\) a derivation of \(R_F\), and for all \(x\in R_F\). Also if \(g(x)^n=0\) for all \(x\in I\) then \(g=0\). The crucial Theorem 5 in the paper is incomplete since the reference at the end of its proof does not apply. A complete argument for this result appears in [\textit{T.-K. Lee}, Commun. Algebra 27, No. 8, 4057-4073 (1999; Zbl 0946.16026)]. This paper of Lee also contains the main results here but for the somewhat more general situation when \(I\) is a dense right ideal of \(R\).
scientific article; zbMATH DE number 2104903