Generalized derivations with nilpotent values on multilinear polynomials.
Let \(K\) be a commutative ring with identity, \(R\) a \(K\)-algebra, \(f(X)\) a multilinear polynomial over \(K\) in noncommuting variables, and with an invertible coefficient, and \(g\) a generalized derivation of \(R\). The authors show that if \(R\) is prime with no nonzero nil right ideal, \(f(X)\) is not central valued on \(R\), and \(g(f(r_i))\) is nilpotent for all \(r_i\in I\), a nonzero ideal of \(R\), then \(g=0\). When \(R\) is semiprime and when \(g(f(r_i))^n=0\) for all \(r_i\in R\), with \(n\) a fixed positive integer, then \([f(X),y]g(z)\) is an identity for \(R\).
scientific article; zbMATH DE number 5119517