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 i...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.展开更多
Let R be a prime ring with center Z, 5 ： R → R a nonzero skew derivation, and n a fixed positive integer. In this paper, we show that R is a commutative ring if （i） [（δ（[x, y]）, [x, y]]n = 0 for all x, y ∈ R ...Let R be a prime ring with center Z, 5 ： R → R a nonzero skew derivation, and n a fixed positive integer. In this paper, we show that R is a commutative ring if （i） [（δ（[x, y]）, [x, y]]n = 0 for all x, y ∈ R or （ii） [（δ（x）, x]n e Z for all x ∈ R, except some specific cases.展开更多
文摘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.
文摘Let R be a prime ring with center Z, 5 ： R → R a nonzero skew derivation, and n a fixed positive integer. In this paper, we show that R is a commutative ring if （i） [（δ（[x, y]）, [x, y]]n = 0 for all x, y ∈ R or （ii） [（δ（x）, x]n e Z for all x ∈ R, except some specific cases.
