Filtrations of modules, the Chow group, and the Grothendieck group

The aim of this paper is to show that a finitely generated module over a Noetherian ring defines a unique cycle class in the components with codimension zero and one of the Chow group of the ring. The main theorem generalizes a classical result over integrally closed domains and implies the isomorphism between the Chow group and the Grothendieck group under certain conditions. We also discuss the difference between the map constructed in this paper and the Riemann-Roch map.