On generalized (α, β)-derivations of semiprime rings

We investigate some properties of generalized (alpha,beta)-derivations on semiprime rings. Among some other results, we show that if g is a generalized (alpha,beta)-derivation, with associated (alpha,beta)-derivation delta, on a semiprime ring R such that [g(x),alpha(x)]=0 for all xin R, then delta(x)[y,z]=0 for all x,y,zin R and delta is central. We also show that if alpha,nu,tau are endomorphisms and beta,mu are automorphisms of a semiprime ring R and if R has a generalized (alpha,beta)-derivation g, with associated (alpha,beta)-derivation delta, such that g([mu(x),w(y)])=[nu(x),w(y)]_{alpha,tau}, where w:Rrightarrow R is commutativity preserving, then [y,z]delta(w(p))=0 for all y,z,pin R.