An ideal I of a commutative ring D with identity is called an SFT ideal if there exist a finitely generated ideal J with $J\subseteq I$ and a positive integer k such that $a^k\in J$ for each $a\in I$. We prove that for a non-SFT maximal ideal M of an integral domain D, ht$(M〚X〛/MD〚X〛)\ge 2^{{\mathrm{\aleph }}_1}$ if either (1) D is a 1-dimensional quasi-local domain (in particular D is a 1-dimensional nondiscrete valuation domain) or (2) M is the radical of a countably generated ideal. In other words, if one of the conditions (1) and (2) is satisfied, then there is a chain of prime ideals in $D〚X〛$ with length at least $2^{{\mathrm{\aleph }}_1}$ such that each prime ideal in the chain lies between $MD〚X〛$ and $M〚X〛$. As an application, assuming the continuum hypothesis we show that if D is either the ring of algebraic integers or the ring of integer-valued polynomials on $\mathbb{Z}$, then $\dim D〚X〛=\mathrm{ht}M〚X〛=\mathrm{ht}(M〚X〛/MD〚X〛)=2^{{\mathrm{\aleph }}_1}$ for every maximal ideal M of D.

如果存在有限生成的理想J且具有恒等式的交换环D的理想I称为SFT理想$Ĵ\subseteq \mathrm{一世}$和一个正整数k使得${\mathrm{一个}}^{ķ}\in Ĵ$ 对于每个 $\mathrm{一个}\in \mathrm{一世}$。我们证明对于积分域D的非SFT最大理想M，ht$（\mathrm{中号}〚X〛/\mathrm{中号}d〚X〛）\ge {\mathrm{2个}}^{{\mathrm{\aleph }}_{\mathrm{1个}}}$（1）D是一维拟局部域（特别是D是一维非离散估值域），或者（2）M是可数生成的理想的根。换句话说，如果满足条件（1）和（2）之一，那么在$d〚X〛$ 长度至少 ${\mathrm{2个}}^{{\mathrm{\aleph }}_{\mathrm{1个}}}$ 这样链中的每个主要理想都位于 $\mathrm{中号}d〚X〛$ 和 $\mathrm{中号}〚X〛$。作为应用，假设连续假设，我们证明如果D是代数整数的环或整数值多项式的环$\mathbb{ž}$， 然后 $\mathrm{暗淡}d〚X〛=\mathrm{H\; T}\mathrm{中号}〚X〛=\mathrm{H\; T}（\mathrm{中号}〚X〛/\mathrm{中号}d〚X〛）={\mathrm{2个}}^{{\mathrm{\aleph }}_{\mathrm{1个}}}$每一个极大理想中号的d。