SFT stability via power series extension over Prüfer domains
A commutative ring \(D\) with identity is called an SFT ring if for each ideal \(I\) of \(D\), there exist a natural number \(k\) and a finitely generated ideal \(J \subseteq I\) such that \(a^k \in J\) for all \(a \in I\). Let dim\(D\) denote the (Krull) dimension of \(D\), let \(x_1, \dots , x_n\) be indeterminates over \(D\), and let \(Dx_1, \dots , x_n\) be the power series ring over \(D\). It is well known that if dim\((Dx_1) < \infty\), then \(D\) is an SFT ring [\textit{J. T. Arnold}, Trans. Am. Math. Soc. 177, 299--304 (1973; Zbl 0262.13007)]. Also, if \(D\) is a finite dimensional SFT Prüfer domain, then dim\((Dx_1, \dots , x_n) = n (\dim D) +1\), and hence \(Dx_1, \dots , x_n\) is an SFT ring [\textit{J. T. Arnold}, Indiana Univ. Math. J. 31, 897--911 (1982; Zbl 0475.13014)]. However, the power series ring over an SFT ring need not be an SFT ring [\textit{J. Coykendall}, J. Algebra 256, 85--96 (2002; Zbl 1069.13011)]. In this paper, the authors extend Arnold's result to infinite dimensional SFT Prüfer domains. Let \(D\) be an SFT Prüfer domain. They first show that \(Dx_1, \dots , x_n_{D \setminus (0)}\) is an \(n\)-dimensional (noetherian) regular ring, and then they prove that \(Dx_1, \dots , x_n\) is an SFT ring. They also show that if \(P\) is a nonzero prime ideal of \(D\) with ht\(P = m < \infty\), then \[ \text{ht}(Px_1, \dots , x_n) = \begin{cases} mn &\text{if \(P\) is nonmaximal and \(\bar{D}_{\bar{D} \setminus (0)}x_1, \dots , x_n\)}\\ & \neq \bar{D}x_1, \dots , x_n_{\bar{D} \setminus (0)}, \text{ where \(\bar{D} = D/P\)},\\ (m-1)n + 1 &\text{otherwise.} \end{cases} \]
scientific article; zbMATH DE number 5347175