integral closure of a filtration relativeto a noetherian module

Let m be a noetherian r-module. in this paper we will introduce the integral closure of a filtration f= {in} n≥0 relative to the noetherian module m and prove some related results. the integral closure of a filtration f~= {in} n≥0 relative to m is a filtration and it has an interesting relationship with the integral closure of the filtration f= {i~n} n≥0, where i~n is the image of in under the natural ring homomorphism r → r/(annr(m)) for every n ≥ 0.