Prime rings with hypercommuting derivations on a Lie ideal (Q1962594)

The authors prove a result involving generalized commutators with derivation. Let \(R\) be a prime ring, \(D\) a nonzero derivation of \(R\), and \(L\) a noncentral Lie ideal of \(R\). Set \([a,b]_1=ab-ba\) and for \(k>1\) \([a,b]_k=[[a,b]_{k-1},b]\). If for each \(u\in L\) \([D(u^m),u^m]_n=0\), where \(m=m(u)\geq 1\) and \(n=n(u)\geq 1\), then when \(R\) contains no nonzero nil right ideal, \(R\) must satisfy the standard polynomial identity \(S_4\).