Power series rings over globalized pseudo-valuation domains
Let R be a commutative integral domain with quotient field K. R is called a pseudo-valuation domain (PVD) if each prime ideal P of R has the property that for \(x,y\in K\), xy\(\in P\) implies that \(x\in P\) or \(y\in P\). A PVD is quasi-local and a quasi-local domain (R,M) is a PVD if and only if (M:M) is a valuation domain. R is called a globalized pseudo- valuation domain (GPVD) if for each prime ideal P of R, \(R_ P\) is a PVD and satisfies a certain pullback description. A PVD or Prüfer domain is a GPVD. An integral domain R satisfies the SFT-property if for each ideal A of R, there exists a positive integer k and a finitely generated ideal \(B\subseteq A\) such that \(a^ k\in B\) for each \(a\in A.\) The main result of this paper is that for a GPVD with \(\dim (R)=n\), the following conditions are equivalent: \((1)\quad R\quad has\) the SFT- property, \((2)\quad \dim (Rx)=n+1,\) and \((3)\quad \dim (Rx)<\infty.\) This generalizes the result due to J. Arnold where R is assumed to be a Prüfer domain rather than a GPVD.
Krull dimension of formal power series ring
scientific article; zbMATH DE number 4085785
scientific article