A remark on the spectra of rings with Gabriel dimension
type: Item
instance of: scholarly article
title: A remark on the spectra of rings with Gabriel dimension
zbMATH Open document ID: 0553.13001
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1984-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/107965
review text: If A is a commutative ring with identity, denote by \({\mathcal I}(A)\) the lattice of ideals of A and by Spec(A) the topological space (endowed with the Zariski topology) of prime ideals of A. The aim of this short note is to prove the following result: Let A and B two commutative rings with identity such that there exists an isomorphism \({\mathcal I}(A)\cong {\mathcal I}(B)\) of partially ordered sets. If A has a Gabriel dimension, then the topological spaces Spec(A) and Spec(B) are homeomorphic. This extends a theorem established in a more complicated manner by \textit{C. Năstăsescu} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 19, 87-92 (1974; Zbl 0289.13001)] in the case A is Noetherian.
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 06B15
zbMATH DE Number: 3880841
zbMATH Keywords: homeomorphic spectra
zbMATH Keywords: lattice of ideals
zbMATH Keywords: Spec
zbMATH Keywords: Gabriel dimension
author: Toma Albu
MaRDI profile type: Publication
cites work: Q3320556
cites work: Q4767094
cites work: The Gabriel dimension of a module
cites work: Q4182848
cites work: Q4777422
description: scientific article; zbMATH DE number 3880841