On a pair of \((\alpha,\beta)\)-derivations of semiprime rings. (Q1779978)

Let \(R\) be a ring with center \(Z\). A mapping \(f\colon R\to R\) is called centralizing if \([f(x),x]\in Z\) for all \(x\in R\). If \(\alpha\) and \(\beta\) are mappings of \(R\) into \(R\), an additive mapping \(d\colon R\to R\) is called an \((\alpha,\beta)\)-derivation if \(d(xy)=\alpha(x)d(y)+d(x)\beta(y)\) for all \(x,y\in R\). The main theorem reads as follows: Let \(R\) be a 2-torsion-free semiprime ring and let \(f,g\) be \((\alpha,\beta)\)-derivations of \(R\), where \(\alpha\) and \(\beta\) are centralizing epimorphisms of \(R\). If \(f(x)x+xg(x)=0\) for all \(x\in R\), then \(g(u)[x,y]=f(u)[x,y]=0\) for all \(x,y,u\in R\) and \(f,g\) map \(R\) into \(Z\). A further result explores the consequences of the weaker condition \(f(x)x+xg(x)\in Z\) for all \(x\in R\).