On some properties of derivations on semiprime rings.
Let \(R\) be a semiprime ring with center \(Z(R)\), and let \(f,g\colon R\to R\). Using easy computations, the author obtains some results when \(f\) and \(g\) are generalized inner derivations satisfying simple identities, and proves some technical commutativity results. Some of these are: when \(R\) is 2-torsion free and \(f(r)=ar+rb\) then \(f\) is a reverse derivation if and only if \(f=0\); if \(f\) and \(g\) are inner derivations so that \(f(x)x+xg(x)=0\) for all \(x\in R\) then \(f=g=0\); when \(f(r)=ar+rb\) and \(f(x)x=xf(x)\) for all \(x\in R\) then \(a,b\in Z(R)\); when \(R\) is prime, \(1\in R\), \(\text{char\,}R\neq 2\), and \(xy^2x-yx^2y\in Z(R)\) for all \(x,y\in R\), then \(R\) is commutative.
scientific article; zbMATH DE number 5077533