摘要
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 ∈ Qr ( 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 ∈ Qr ( RC) and w ∈ Qr ( 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.
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 ∈ Qr （ 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 ∈ Qr （ RC） and w ∈ Qr （ 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.
