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Paper   IPM / M / 11185
School of Mathematics
  Title:   Semistar-Krull and valuative dimension of integral domains
  Author(s):  P. Sahandi
  Status:   Published
  Journal: Ricerche mat.
  Vol.:  58
  Year:  2009
  Pages:   219-242
  Supported by:  IPM
  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 ∗-dimv(D), to be the maximal rank of the ∗-valuation overrings of D. We show that ∗-dimv(D)=n if and only if ∗[X1,…,Xn]-dimv(D[X1,…,Xn])=2n, and that if ∗-dimv(D) < ∞ then ∗[X]-dimv(D[X])=∗-dimv(D)+1. In general ∗-dim(D) ≤ ∗-dimv(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)=∗-dimv(D). As an application, ∗-quasi-Prüfer 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 wD-Jaffard domain.


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