Abstract
Kishor Shah proved the existence and uniqueness of a chain of ideals between the ideals $ I $ the ring $ R $ and the integral closure of the same, these ideals preserve the first $ k +1$ the Hilbert coefficients of $I$, i.e., there are ideals $I_k $ containing the ideal $I$, for every integer $ k = 1, \ ldots, d $ such that $ I \subset I_ d \subset \ldots \subset I_1 \subset \overline {…