Abstract
In this paper, we assume that M is Γ-ring then we will present the definition of symmetric Γ-n-centralizers and the definition of Jordan Γ-n-centralizers of M. After that, we try to find the commutativity of a prime Γ-ring that has a left Γ-n-centralizers of M when it satisfies certain identities. Finally, we find the relationship between Jordan Γ-n-centralizers and symmetric Γ-n-centralizers when M is semiprime Γ-ring.
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