Multiplicities of ideals in Noetherian rings (Q1380829)

Let \(R\) be a commutative Noetherian ring. For any ideal \(I\) of \(R\) there is defined a multiplicity function \(\text{mult}_I: \text{Spec} (R)\to \mathbb{N}\) by assigning to each prime ideal \({\mathfrak p}\) of \(R\) the multiplicity \(\text{mult}_I ({\mathfrak p})\) of \({\mathfrak p}\) at \(R/I\). The aim of this paper is to characterize all the functions \(\psi: \text{Spec} (R)\to \mathbb{N}\) which arise as \(\text{mult}_I\) for some ideal \(I\) of \(R\).