Generalized derivations in prime rings

Abstract

Let R be a ring, a,b ∈ R, (D,α) and (G,β) be two generalized derivations of R. It is proved that if aD(x) = G(x)b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b is contained in C, then α = β = 0 and there exist p,q∈Q _r (RC) such that D(x) = px and G(x) = qx for all x∈R; (ii) If both a and b are contained in C, then either a = b = 0 or D and G are C -linearly dependent; (iii) If neither a nor b is contained in C, then there exist p,q∈Q _r (RC) and w∈Q _r (R) such that α(x) = [q,x]_and β(x) = [x,p]_for all x∈R, whence D(x) = wx−xq and G(x) = xp + avx with v∈ C and aw−pb = 0.