Generalized derivations and left multipliers on Lie ideals.
Let \(R\) be a prime ring with \(\text{char\,}R\neq 2\), extended centroid \(C\), Utumi quotient ring \(U\), noncentral Lie ideal \(L\), and generalized derivations \(F\) and \(D\). For fixed positive integers \(n\) and \(m\) let \(P(x)\) be the expression \(F(x^{n+m+1})-F(x)x^{n+m}-x^nF(x)x^m\). The main result in the paper assumes that \(P(x)\) is an identity for \(L\) and shows that either \(D=0\) and \(F(r)=ar\) for some \(a\in U\) and all \(r\in R\), or else \(R\) satisfies the standard polynomial identity \(s_4\) of degree four. The first possibility must hold when \(P(x)\) is an identity for \(R\). When \(R\) satisfies \(s_4\) and \(P(x)\) is an identity for \(L\), then \(n+m\) even forces \(D=0\), and \(n+m\) odd forces \(D\) to be a derivation and there is \(a\in U\) so that for all \(r\in R\), \(F(r)=ar-D(r)\) when \(n\) is even and \(F(r)=ar+D(r)\) when \(n\) is odd.
scientific article; zbMATH DE number 5909092
scientific article