Krull dimension of power series rings over non-SFT domains
type: Item
instance of: scholarly article
title: Krull dimension of power series rings over non-SFT domains
zbMATH Open document ID: 1390.13012
author: Phan Thanh Toan
author: B. G. Kang
published in: Journal of Algebra
publication date: 2018-03-01T00:00:00Z
review text: 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 \(D\in {\mathcal C}\).
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 6845900
zbMATH Keywords: Krull dimension
zbMATH Keywords: non-SFT domain
zbMATH Keywords: power series ring
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2017.12.011
OpenAlex ID: W2789746955
cites work: Krull Dimension in Power Series Rings
cites work: Power series rings over Prüfer domains
cites work: Q3931523
cites work: The SFT property does not imply finite dimension for power series rings.
cites work: Integer-valued polynomials
cites work: FORMAL POWER SERIES RINGS OVER ZERO-DIMENSIONAL SFT-RINGS
cites work: The Krull dimension of power series rings over almost Dedekind domains
cites work: Q3665229
cites work: Generalized Dedekind domains and their injective modules
cites work: On the prime ideals of the ring of entire functions
cites work: Q5466551
cites work: Q5661989
cites work: Q3270120
cites work: Q5603302
cites work: The Krull dimension of power series rings over non-SFT rings
cites work: A localization of a power series ring over a valuation domain
cites work: On Močkor's question
cites work: SFT stability via power series extension over Prüfer domains
cites work: Krull dimension of mixed extensions
cites work: Krull-dimension of the power series ring over a nondiscrete valuation domain is uncountable
cites work: Krull dimension of a power series ring over a valuation domain
cites work: Constructing chains of primes in power series rings
cites work: CONSTRUCTING CHAINS OF PRIMES IN POWER SERIES RINGS, II
cites work: Arnold's Theorem on the Strongly Finite Type (SFT) Property and the Dimension of Power Series Rings
cites work: A note on the dimension theory of rings
cites work: On the dimension theory of rings. II
DOI: 10.1016/J.JALGEBRA.2017.12.011
description: scientific article; zbMATH DE number 6845900