Generalized Higher Left Centralizer of Prime Γ-Rings

In this paper we introduce the concepts of generalized higher left centralizer and generalized Jordan higher left centralizer of  Γ-rings M as well as we proved that every generalized Jordan higher left centralizer of certain Γ-ring M is generalized higher left centralizer of M and we prove every Jordan generalized higher left centralizer of certain  Γ-ring M is Jordan generalized triple higher left centralizer of M.