Integral dependence over a filtration

Abstract

In this note, we introduce in a commutative unitary ring the notions of integral and strongly integral filtration over another one, which generalize the notion of reduction of an ideal introduced by Rees and Northcott. We study the integral dependence for regular and E. P. filtrations and obtain as consequence that the prüferian closure of a filtration in a noetherian ring is a semi prime operation. We prove that if the filtration g is integral over f and if ν̄ g (resp. ν̄ f) is the homogenous pseudo-valuation associated with g (resp. with f), then ν ̄ g≤ ν ̄ f and the converse is true if f is a regular filtration on a noetherian ring. In the same way, we show that for two filtrations f and g in a noetherian ring such that f≤ g, if the multiplicity e f ( A) of f exists and if g satisfies the asymptotic formula for multiplicities, then e f ( A) = e g ( A) if g is integral over f and the converse property is true for any regular filtration f in an M-ring. The paper is closed by proving several characterizations of E.P. filtrations in integrally closed Nagata domains.