In this Paper we show the following: Let R be a prime ring (with characteristic different two) and a $in$ R. Let Θ, $phi$ : R longrightarrow R be automorphisms and let d : R longrightarrow R be a nonzero Θ-derivation. (i) if［d($chi$), a］Θo$phi$ = 0 (or d(［$chi$, a］$phi$ = 0) for all $chi$ $in$ R, then a+$phi$(a) $in$ Z, the conte. of R, (ii) if〈d($chi$), a〉 = 0 for all $chi$$in$R, then d(a) =0. (iii) if [ad($chi$), $chi$］$phi$= 0 for all $chi$$in$R, then either a = 0 or R is commutative.