Commutative Ring Theory

8 The Generalized Samuel Numbers

H. DICHI Département de Mathématiques, Université d’Abidjan, Abidjan, Côte d’ivoire

1. Introduction

Throughout this note, A will denote a commutative ring with unit element. For two filtrations $\text{f}=\left({\text{I}}_{\text{n}}\right)$ and $\text{g}=\left({\text{J}}_{\text{n}}\right)$ on A, we put ${\overline{\text{v}}}_{\text{f}}(\text{g})=\underset{\text{n}-->\infty }{\lim }\frac{{\text{v}}_{\text{f}}{\text{(J}}_{\text{n}}\text{)}}{\text{n}}$ and ${\overline{\text{w}}}_{\text{f}}(\text{g})=\underset{\text{n}-->\infty }{\lim }\frac{{\text{w}}_{\text{f}}{\text{(J}}_{\text{n}}\text{)}}{\text{n}}$, where for an ideal J on A, ${\text{v}}_{\text{f}}\text{(J})=\sup \left\{\text{n};\text{J}\subseteq {\text{I}}_{\text{n}}\right\}$ and ${\text{w}}_{\text{f}}(\text{J})=\inf \{\text{n};{\text{I}}_{\text{n}}\subseteq \text{J}\}$. A filtration f = (I_n) on A is a decreasing sequence of ideals I_n such that I₀ = A and ${\text{I}}_{\text{n}}{\text{I}}_{\text{m}}\subseteq {\text{I}}_{\text{n+m}}$ for all integers n,m. The set $\mathbb{F}(\text{A})$ of filtrations on the ring A is ordered by $\text{f}=\left({\text{I}}_{\text{n}}\right)\le \text{g}=\left({\text{J}}_{\text{n}}\right)\text{if}{\text{I}}_{\text{n}}\le {\text{J}}_{\text{n}}$ for all n. The numbers ${\overline{\text{v}}}_{\text{f}}\text{(g})$ and ${\overline{\text{w}}}_{\text{f}}\text{(g)}$ are called the generalized Samuel numbers of the filtrations f and g. They have been studied in [10] by Samuel in the particular case when ...