Abstract: We prove that the Krull dimension of the power series ring over a nonSFT domain is at least $\aleph_2$. In particular, the Krull dimension of the power series ring over the ring of algebraic integers is $\aleph_2$ and the height of $P $ is $\aleph_2$ as well for each nonzero prime ideal $P$ of the ring of algebraic integers under the Continuum Hypothesis.
A ring $D$ is called an SFT ring if for each ideal $I$ of $D$, there exists a finitely generated ideal $J$ of $D$ with $J \subseteq I$ and a positive integer $k$ such that $a^k \in J$ for all $a \in I$. A ring is nonSFT if it is not SFT.
For a cardinal number a and a ring $D$, we say that dim$(D) \geq a$ if $D$ has a chain of prime ideals with length $\geq a$. J. T. Arnold showed that if $D$ is a non-SFT ring then dim$(D) \geq \aleph_0$. Let $\mathscr{C}$ be the class of non-SFT domains.
The class $\mathscr{C}$ includes the class of finite-dimensional non-discrete valuation domains, the class of non-Noetherian almost Dedekind domains, the class of completely integrally closed domains that are not Krull domains, the class of integral domains with non-Noetherian prime spectrum, and the class of integral domains with a nonzero proper idempotent ideal. The ring of algebraic integers, the ring of integer-valued polynomials on $\mathbb{Z}$ and the ring of entire functions are also members of the class $\mathscr{C}$. In this talk we prove that dim$(D) \geq 2^{\aleph_1}$ for every $D \in \mathscr{C}$ and that under the continuum hypothesis $2^{\aleph_1}$ is the greatest lower bound of dim$(D)$ for $D \in \mathscr{C}$. On the other hand there exists a (finite dimensional) SFT domain $D$ such that dim$(D) \geq 2^{\aleph_1}$.
