Integral closure of a filtration relative to a module

In this paper, the concept of integral dependence over a filtration f on a ring A, relative to a module M is introduced and several properties concerning the integral closure ClosA(f, M) of a pair (f, M) are established. Specially, we give a complete description of ClosA(f, M) when A is a noetherian ring of Krull dimension at most 1. The paper is closed by giving a constructive method to express the integral closure of a pair (f, A) when f is strongly AP an a noetherian integral domain of Krull dimension 1.