Mode Of The Relative Importance Of The Number

The notion multiplicity is a very important notion in commutative algebra and algebra ge-ometry. It may be used to study a lot of properties of commutative rings. One of the importantproperties of the multiplicity is that it has the additive property with respect to a short exact se-quence. Hence it can be thought as an extension of the notion of length of an Artinian module.The multiplicity is defined by means of Samuel polynomials, where the length function plays animportant role.In this paper we will use the multiplicity instead of the length function to introduce a newfunction, which has the additive property. We will call this function the relative multiplicity.Some of the properties of such multiplicity will be given in the paper.Our method is as follows. For a graded ring R =(?)Rn which is finitely generated over aNoetherian local ring R0 and a finitely generated graded R?module M =(?)Mn, we will use thenotion of multiplicity to define a new Poincare series and show that this series is a rational function.Then we will define relative multiplicity similar as the definition of multiplicity. Moreover, wewill study the properties of this new multiplicity which is similar to the multiplicity.