Annales Mathematiques Africaines

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Contenu > Anciens Numéros > Volume_2

Generalized Samuel Numbers MATH and MATH
on a Semi-Module

Université de Cocody, IRMA
08 BP 2035 Abidjan 08, Côte d'Ivoire

Eric Dago AKEKE
Université de Cocody, UFRMI
22 BP 582 Abidjan 22, Côte d'Ivoire

Philippe AYEGNON
E.N.S. d'Abidjan 08 BP 10 Abidjan 08, Côte d'Ivoire

Mathematics Subject Classifications: 13A15, 13A18.
Key words : ring, semi-ring, semi-module, filtration.


The Asymptotic Theory of ideals originated with the investigations in a nœtherian ring $A$ of the Samuel number $\overline v_{I}(J)$ associated with each pair $(I,J)$ of nonnilpotent ideals having the same radical and
the limit being reached from below and MATH.
The number $\overline w_{I}(J)$ is defined in a symmetric situation. In AYEGNON - D. SANGARE these numbers have been defined for pairs $(f,g)$ where MATH and MATH are filtrations on a ring $A$. In this paper we extend these definitions to generalized Samuel numbers MATH and MATH where MATH and MATH are filtrations on the $A$-semi-module $M$ by setting :
if these limits exist in MATH.

It is shown that if $\varphi$, $\theta$ and $\eta$ are filtrations on the $A$-semi-module $M$ such that $\varphi$ is a valuative reduction of $\theta$ then MATH

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