Krull dimension of power series rings over non-SFT domains
All rings considered in this paper are commutative with identity. The authors extend the concept of Krull dimension of a ring \(R\) as follows. For a cardinal number \(\alpha\), we say that \(\dim (R)=\alpha\) if \(R\) has a chain of prime ideals with length \(\alpha\) but no longer chains. Recall that an ideal \(I\) of a ring \(R\) is said to be an SFT-ideal if there exist a finitely generated ideal \(J\subseteq I\) and an integer \(k\geq 1\) such that \(a^k\in J\) for all \(a\in I\). We say that \(R\) is an SFT-ring if all its ideals are SFT. The class \({\mathcal C}\) of non-SFT domains include finite dimensional nondiscrete valuation domains, non-Noetherian almost Dedekind domains, completely integrally closed domains that are not Krull domains, integral domains with non-Noetherian prime spectrum, integral domains with a nonzero proper idempotent ideal, the ring \({\mathcal O}\) of algebraic integers, the ring \(\mathrm{Int}(\mathbb{Z})\) of integer-valued polynomials on \(\mathbb{Z}\) and the ring \({\mathcal E}\) of entire functions. In this paper, the authors prove that \(\dim(DX)\geq 2^{\aleph_0}\) for every \(D\in {\mathcal C}\) and under the continuum hypothesis, \(2^{\aleph_1}\) is the greatest lower bound of \(\dim(DX)\) for \(D\in {\mathcal C}\).
scientific article; zbMATH DE number 6845900
scientific article