Commuting and centralizing generalized derivations on Lie ideals in prime rings.
Let \(R\) be a noncommutative prime ring with \(\text{char\,}R\neq 2\), Utumi quotient ring \(U\), extended centroid \(C\), and noncentral Lie ideal \(L\). For \(x,y\in R\) set \([x,y]_1=xy-yx\) and for \(k\geq 1\) let \([x,y]_{k+1}=[[x,y]_k,y]_1\). The main result assumes that \(F\) and \(G\) are generalized derivations of \(R\) so that for a fixed integer \(k\geq 1\) and all \(x\in L\), \([F(x),x]_kx=x[G(x),x]_k\). The authors prove that for all \(x\in R\), \(F(x)=xa\) and \(G(x)=(a+z)x\) for some \(a\in U\) and \(z\in C\), or else \(R\) satisfies the standard identity \(S_4\) and one of the following holds: a) either \(F\) and \(G\) are inner with closely related constants, or b) for all \(x\in R\), \(F(x)=ax+d(x)\) and \(G(x)=bx+cx-d(x)\) for \(a,b,c\in U\) with \(a+b-c\in C\) and some derivation \(d\) of \(U\). When \(L=R\) the first conclusion, \(F(x)=ax\) and \(G(x)=(a+z)x\), must hold.
scientific article; zbMATH DE number 5980718
scientific article