Contents

**
Filtrations, analytic spread, and integral dependence
**

**Dramane Bio SALIFOU
**

107 31 Ave NE, Calgary, Alberta, T2E 2E8, Canada

*Mathematics Subject Classification:* **(2010) ** 13A30, 13E05

*Key words: *nœtherian filtration, Rees ring, dimension, integral dependence,
analytic spread.

Abstract:

Given a positive integer k which may be infinite, an ideal J of a
commutative ring A and a filtration f of A, the purpose of this note is firstly, to
prove that the (Krull) dimension of the generalized Rees quotient ring
, where ℜ(A,f) is the generalized Rees quotient ring of the filtration
f, and that is denoted by λ_{J}(f,k), extends the usual Analytic Spread of a
nœtherian filtration f of a nœtherian ring A. Secondly, if g is another
filtration of the nœtherian ring A which is integral over f, then the equality
λ_{J}(g,k) = λ_{J}(f,k) holds.