Prime Gamma Rings with Centralizing and Commuting Generalized Derivations

Let $M$ be a prime $Γ$-ring satisfying a certain assumption and $D$ a nonzero derivation on $M$. Let $f:M\rightarrow M$ be a generalized derivation such that $f$ is centralizing and commuting on a left ideal $J$ of $M$. Then we prove that $M$ is commutative.

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