A note on unions of ideals and cosets of ideals
type: Item
instance of: scholarly article
title: A note on unions of ideals and cosets of ideals
zbMATH Open document ID: 0827.13001
author: Michael Zandt
published in: Beiträge zur Algebra und Geometrie
publication date: 1995-06-25T00:00:00Z
full work available at URL: https://eudml.org/doc/224943
full work available at URL: no.1
review text: Let \(A\) be a commutative ring with 1, let \(M\) be an arbitrary \(A\)-module. Let \(m\text{-spec} (A)\) be the set of all maximal ideals of \(A\), \(m \text{-supp} (M) = m\text{-spec} (A) \bigcap \text{supp} (M)\) and \(N\) be the set of all natural numbers. With this notation, the author establishes the following results: Result 1. Let \(I\), \(I_1, \ldots, I_r\) be ideals, \(I_{r + 1}, \ldots, I_{r + s}\) prime ideals of \(A\), where \(r,s \in N\), \(r + s \geq 1\). If \(r \leq \min \{\text{card} (A/{\mathfrak m}:{\mathfrak m}\in m\text{-spec} (A)\}\), then \(I \subset \bigcup^{r + s}_{i = 1} I_i\) implies \(I \subset I_i\) for some \(i\) with \(1 \leq i \leq r + s\). Result 2. Let \(I\), \(I_1, \ldots, I_t\) be ideals of \(A\), \(t \in N\), \(a_{ij} \in A\), \(j = 1, \ldots, n_i\), \(n_i \in N\) for \(i = 1, \ldots, t\). Suppose \(I \not \subset I_i\) for all \(i\) and \(\sum^t_{i = 1} n_i < \inf \{\text{card} (A/{\mathfrak m}) :{\mathfrak m}\in m\text{-spec} (A)\}\). Then \(I \not \subset \bigcup^t_{i = 1} \bigcup^{n_i}_{j = 1} (a_{ij} + I_i)\).
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 767415
zbMATH Keywords: prime ideals
zbMATH Keywords: maximal ideals
reviewed by: K. Chandrasekhara Rao
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 767415