In this paper,  we introduce the concepts of  higher reverse left (resp.right)   centralizer, Jordan higher reverse left (resp. right) centralizer, and Jordan triple higher reverse left (resp. right) centralizer of  G-rings. We prove that every Jordan higher reverse left (resp. right) centralizer of a 2-torsion free prime G-ring M is a higher reverse left (resp. right) centralizer of  M.

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In this study we introduced the concepts of generalized higher reverse left (accordingly, right) centralizer, and Jordan generalized higher reverse left (accordingly, right) centralizer of rings. The definition of Jordan triple generalized higher reverse left (accordingly, right) centralizer was deduced. The most important findings of this paper are as follows: every Jordan generalized higher reverse left (accordingly right) centralizer of a 2-torsion free prime ring R into itself is a generalized higher reverse left (accordingly right) centralizer of R. The results have confirmed that every Jordan generalized higher reverse left (accordingly right) centralizer is a generalized higher reverse left (accordingly right) centralizer within certain conditions.