Commuting derivations of semiprime rings. (Q2908912)

A generalized derivation on the ring \(R\) is an additive map \(D\colon R\to R\) such that \(D(xy)=D(x)y+xd(y)\) for all \(x,y\in R\), where \(d\) is a derivation on \(R\), called the associated derivation. In the context of domains (not necessarily with 1), the authors attempt to show that certain conditions on \(D\) imply that \([d(x),x]=0\) for all \(x\in R\) or \([d(x),x^2]=0\) for all \(x\in R\).NEWLINENEWLINE Most of the proofs are incorrect, the most common error being cancellation of elements which are not necessarily nonzero.