“School of Mathematics”
Back to Papers HomeBack to Papers of School of Mathematics
Paper IPM / M / 11185  


Abstract:  
Given a stable semistar operation of finite type ∗ on
an integral domain D, we show that it is possible to define in a
canonical way a stable semistar operation of finite type ∗[X]
on the polynomial ring D[X], such that, if n:=∗dim(D),
then n+1 ≤ ∗[X]dim(D[X]) ≤ 2n+1. We also
establish that if D is a ∗Noetherian domain or is a
Prüfer ∗multiplication domain, then
∗[X]dim(D[X])=∗dim(D)+1. Moreover we
define the semistar valuative dimension of the domain D, denoted
by ∗dim_{v}(D), to be the maximal rank of the
∗valuation overrings of D. We show that
∗dim_{v}(D)=n if and only if
∗[X_{1},…,X_{n}]dim_{v}(D[X_{1},…,X_{n}])=2n, and that if
∗dim_{v}(D) < ∞ then
∗[X]dim_{v}(D[X])=∗dim_{v}(D)+1. In general
∗dim(D) ≤ ∗dim_{v}(D) and equality holds if D is
a ∗Noetherian domain or is a Prüfer
∗multiplication domain. We define the ∗Jaffard domains
as domains D such that ∗dim(D) < ∞ and
∗dim(D)=∗dim_{v}(D). As an application,
∗quasiPrüfer domains are characterized as domains D
such that each (∗,∗′)linked overring T of D, is a
∗′Jaffard domain, where ∗′ is a stable semistar
operation of finite type on T. As a consequence of this result we
obtain that a Krull domain D, must be a w_{D}Jaffard domain.
Download TeX format 

back to top 