Generalized derivations with annihilating and centralizing Engel conditions on Lie ideals.
Let \(R\) be a prime ring with center \(Z(R)\), extended centroid \(C\), right Utumi quotient ring \(U\), noncentral Lie ideal \(L\), and generalized derivation \(F\). For \(a,b\in R\) set \([a,b]_1=ab-ba\) and for \(k>1\) let \([a,b]_k=[[a,b]_{k-1},b]_1\). Assume that for some nonzero \(a\in R\) and positive integer \(k\), \(a[F(x),x]_k=0\) for all \(x\in L\). The first main result of the author shows that for all \(x\in R\), \(F(x)=cx\) for some \(c\in C\) or \(R\) satisfies the standard polynomial identity \(S_4\) and either \(\text{char\,}R=2\) or there are \(c\in C\) and \(q\in U\) so that \(F(x)=qx+xq+cx\). This theorem is used to prove the more general statement that assumes \(a[F(x),x]_k\in Z(R)\) for all \(x\in L\) and has the same conclusion as the first result.
scientific article; zbMATH DE number 6121271
scientific article