Let ܯ be a prime-ring with 2-torsion free, ܫ a nonzero ideal of M and ݂: ܯ → ܯ a left generalized derivation of ,ܯ with associated nonzero derivation d on .ܯ If )ݔ(݂ ∈ )ܯ(ܼ for all ݔ ∈ ,ܫ then ܯ is a commutative-ring.
Let be a prime Γ-ring satisfying a certain assumption and a nonzero derivation on M. Let : → be a generalized Jordan derivation such that is centralizing and commuting on a left ideal of. Then we prove that is commutative.
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