Nilpotent and invertible values in semiprime rings with generalized derivations.
The authors consider a ring \(R\) with a nonzero right ideal \(I\), a generalized derivation \(F\) of \(R\), and for a fixed positive integer \(n\) and all \(x,y\in I\), the elements \(H(x,y)=(F(xy)-yx)^n\in R\). They prove that when \(I=R\) is semiprime and all \(H(x,y)\) are either nilpotent or invertible then \(R\) is a division ring \(D\) or \(R\cong M_2(D)\). When \(R\) is a prime ring and all \(H(x,y)=0\) then \([I,I]I=0\) and \(F(x)=ax+xb\) for \(a,b\in R\) with \((a-\alpha)I=0\) and \((b-\beta)I=0\) for \(\alpha,\beta\in C\), the extended centroid of \(R\). This latter result requires the authors to first prove that \(R\) satisfies a nontrivial generalized polynomial identity.
scientific article; zbMATH DE number 5956215