A theorem on derivations in semiprime rings (Q1915567)

By means of an elementary but clever calculation, the author proves a theorem on the Engel condition with a derivation. Set \([x,y]_1=xy-yx\) and for \(k>1\), \([x,y]_k=[[x,y]_{k-1},y]_1\). The result proved is: If \(R\) is an \((n+1)!\)-torsion free semiprime ring, \(L\) is a nonzero left ideal of \(R\), and \(D\) is a nonzero derivation of \(R\) so that for all \(x\in L\), \([D(x),x]_n=0\), then \(D(L)=0\) or \(R\) contains a nonzero central ideal.