On \(n\)-absorbing ideals and the \(n\)-Krull dimension of a commutative ring
type: Item
instance of: scholarly article
zbMATH Open document ID: 1355.13005
author: Hosein Fazaeli Moghimi
author: Sadegh Rahimi Naghani
published in: Journal of the Korean Mathematical Society
publication date: 2016-12-02T00:00:00Z
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 6658981
zbMATH Keywords: \(n\)-absorbing ideal
zbMATH Keywords: \(n\)-Krull dimension
zbMATH Keywords: \(n\)-height
zbMATH Keywords: Artinian ring
zbMATH Keywords: Dedekind domain
MaRDI profile type: Publication
full work available at URL: jkms.j150072
OpenAlex ID: W2560005514
DOI: 10.4134/JKMS.J150072
title: On \(n\)-absorbing ideals and the \(n\)-Krull dimension of a commutative ring
review text: Let \(R\) be a commutative ring and \(n\) a positive integer. In the paper under review, the authors introduce the \(n\)-Krull dimension of \(R\), denoted by \(\dim_n R\), which is the supremum of the lengths of chains of \(n\)-absorbing ideals of \(R\). Recall that for a positive integer \(n\), a proper ideal \(I\) of \(R\) is called an \(n\)-absorbing ideal if whenever \(x_1\cdots x_{n+1}\in I\) for \(x_1,\ldots,x_{n+1} R\), then there are \(n\) of the \(x_i\)'s whose product is in \(I\). It is clear that 1-absorbing ideals are just prime ideals. The authors extend several results of Krull dimension to \(n\)-Krull dimension and study the \(n\)-Krull dimension in several classes of commutative rings. For example, the \(n\)-Krull dimension of an Artinian ring is finite for every positive integer \(n\). In particular, if \(R\) is an Artinian ring with \(k\) maximal ideals and \(l(R)\) is the length of a composition series for \(R\), then dim\(_n R = l(R)-k\) for some positive integer \(n\). It is proved that a Noetherian domain \(R\) is a Dedekind domain if and only if dim\(_n R=n\) for every positive integer \(n\) if and only if dim\(_2 R=2\). It is also shown that Krull's (Generalized) Principal Ideal Theorem does not hold in general when prime ideals are replaced by \(n\)-absorbing ideals for some \(n > 1\).
reviewed by: Parviz Sahandi
description: scientific article; zbMATH DE number 6658981