On generalized derivations of prime rings. (Q815163)

Let \(R\) be a ring with center \(Z(R)\); and for \(x,y\in R\), denote by \([x,y]\) and \(x\circ y\) respectively the commutator \(xy-yx\) and the anti-commutator \(xy+yx\). An additive endomorphism \(F\) of \(R\) is called a generalized derivation if there exists a derivation \(d\) on \(R\) such that \(F(xy)=F(x)y+xd(y)\) for all \(x,y\in R\), and \(d\) is called the associated derivation of \(F\). Now let \(R\) be a prime ring with \(\text{char}(R)\neq 2\), and \(I\) a nonzero ideal of \(R\); and let \(F\) be a generalized derivation with nonzero associated derivation \(d\). It is shown that \(R\) must be commutative if one of the following holds for all \(x,y\in I\): (i) \(d(x)\circ F(y)=0\); (ii) \([d(x),F(y)]=0\); (iii) \(d(x)\circ F(y)=x\circ y\); (iv) \(d(x)\circ F(y)+x\circ y=0\); (v) \(d(x)F(y)-xy\in Z(R)\); (vi) \(d(x)F(y)+xy\in Z(R)\); (vii) \([d(x),F(y)]=[x,y]\); (viii) \([d(x),F(y)]+[x,y]=0\). These results generalize known results for derivations.