Buchsbaumness of certain generalization of the associated graded modules in the equi-I-invariant case

The present article introduces a generalization of the associated graded modules, writing Gα(a,E). Suppose that the equi-I-invariant case occurs. Then it is shown that our graded module Gα(a,E) is Buchsbaum over R(a) for any α>0, if the given A-module E is so over A. Moreover, it is also shown that E is Buchsbaum over A iff Gα(m,E) is so over R(m) for all α>0, where m is the maximal ideal of A.