On Derivations in Semiprime Rings

Let R be a ring, S a nonempty subset of R and d a derivation on R . A mapping is called commuting on S if [ f ( x ), x ] = 0 for all x ∈ S . In this paper, our purpose is to produce commutativity results for rings and show that if R is a 2-torsion free semiprime ring and I a nonzero ideal of R , then a derivation d of R is commuting on I if one of the following conditions holds: (i) d ( x ) ∘ d ( y ) = x ∘ y (ii) d ( x ) ∘ d ( y ) = − ( x ∘ y ) (iii) d ( x ) ∘ d ( y ) = 0 (iv) [ d ( x ), d ( y )] = − [ x , y ] (v) d ( x ) d ( y ) = xy (vi) d ( x ) d ( y ) = − xy (vii) d ( x ) d ( y ) = yx (viii) d ( x ) d ( x ) = x 2 for all x , y ∈ I . Further, if d ( I ) ≠ 0, then R has a nonzero central ideal. Finally, some examples are given to demonstrate that the restrictions imposed on the hypotheses of the various results are not superfluous.