Power values of generalized derivations with annihilator conditions in prime rings.
Let \(R\) be a prime ring with Utumi quotient ring \(U\) and noncommutative Lie ideal \(L\). A generalized derivation of \(R\) is \((H,d)\) for additive \(H,d\colon R\to R\) where \(d\in\text{Der}(R)\) and for all \(x,y\in R\), \(H(xy)=H(x)y+xd(y)\). Let \(0\neq a\in R\) and fix nonnegative integers \(s\) and \(t\), and \(n\geq 1\). In the main result of the paper the authors assume that for all \(y\in L\), \(a(y^sH(y)y^t)^n=0\) and show that \(s=0\) and that for all \(r\in R\), \(H(r)=br\) with \(ab=0\), except when \(R\) embeds in \(M_2(F)\) for a field \(F\). In this latter case, either \(\text{char\,}R=2\) or there is \(b\in U=M_2(F)\) so \(H(r)=rb\) or \(br\) (depending on the parities of \(s\) and \(t\)) and when \(s\) and \(t\) have the same parity \(ab=0\), and when \(s\) and \(t\) have different parity \(ab^n=0\). The major part of the paper is considerable computation in the \(R=M_2(F)\) case.
scientific article; zbMATH DE number 6147176
scientific article