Turkish Journal of Mathematics






Let R be a ring, X\neq (0) an R-bi-module, d: R\ra X a(\sigma,\tau)- derivation with module value such that d\sigma=\sigma d, d\tau=\tau d and U\neq (0) an ideal of R. Furthermore the following properties are also satisfied. \begin{eqnarray*} && \mbox{For }x\in X, a\in R\quad x Ra=0 \mbox{ implies } x=0 \mbox{ or } a=0 \ldots\ldots (G_{1})\\ && \mbox{For }a\in R, x\in X \quad a Rx=0 \mbox{ implies } a=0 \mbox{ or } x=0 \ldots\ldots (G_{2}) \end{eqnarray*} \noindent In this paper we have proved the following results; (1) If (G_{1}) (or (G_{2})) is satisfied and for a \in R, d(U) a=0 (or a d(U)=0) then d=0 or a=0 (2) If (G_{1}) is satisfied and [X,U] \subset C(X) or [X,U]_{\sigma,\tau}\subset C_{\sigma,\tau}(X) then R is commutative (3) Let X be a 2-torsion free R-bi module, d_{1}: R \ra X a(\sigma,\tau)-derivation, d_{2}:R\ra R a derivation such that d_{2}(U) \subset U. If (G_{1}) is satisfied and d_{1}d_{2}(U)=0 then d_{1}=0 or d_{2}=0 (4) Let X be a 2-torsion free R-bi-module. If (G_{1}) and (G_{2}) are satisfied and for a\in U, [d(U), a]_{\sigma, \tau}\subset C_{\sigma,\tau}(X) then a\in Z or d=0.

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