Generalized derivations of prime rings on multilinear polynomials with annihilator conditions. (Q2839025)

Let \(R\) be a prime ring with \(\text{char\,}R\neq 2\), right Utumi quotient ring \(U\), extended centroid \(C\), nonzero right ideal \(T\), and nonzero multilinear polynomial \(f(X)\in C\{X\}\). Set \(xy-yx=[x,y]\). A generalized derivation of \(R\) is an additive map \(g\colon R\to R\) so that \(g(xy)=g(x)y+xd(y)\) for all \(x,y\in R\) and some \(d\in\text{Der}(R)\).NEWLINENEWLINE The authors generalize some results for derivations to generalized derivations. The main result assumes that for some \(a\in R\) with either \(aT\neq 0\) or \(ag(T)\neq 0\), and some generalized derivation \(g\) of \(R\), the expression \(ag(f(X))f(X)\) is an identity for \(T\). The conclusion is that either \([f(x_1,\dots,x_n),x_{n+1}]x_{n+2}\) is an identity for \(T\), or \(g\) is inner with \(g(x)=bx+[c,x]\) for \(b,c\in U\) satisfying \([c,T]T=0=abT\) or \(aT=0=a(b+c)T\). The proof, by steps, proves the theorem in matrix rings when \(g\) is inner, then for \(R\) when \(g\) is inner, and finally for arbitrary \(g\).