Higher order \(z\)-ideals of special rings
type: Item
instance of: scholarly article
title: Higher order \(z\)-ideals of special rings
zbMATH Open document ID: 1484.13007
author: Ahmed Maatallah
author: Ali Benhissi
published in: Beiträge zur Algebra und Geometrie
publication date: 2022-03-22T00:00:00Z
review text: Let \(R\) be a commutative unitary ring. An ideal \(I\) of \(R\) is called \(z\)-ideal ``if whenever any two elements of \(R\) are contained in the same set of maximal ideals and \(I\) contains one of them, then it also contains the other one.'' If \((A_n)_{n\geq 0}\) is an increasing sequence of commutative rings, \(\mathcal{A}X\) is defined by \(\mathcal{A}X=\{\sum_{n\geq 0}a_nX^n\mid a_n\in A_n, \forall n\in\mathbb{N}\}\). A relation between the \(z\)-ideals of \(A_0\) and those of the ring \(\mathcal{A}X\) is given. If \(R\) is a commutative ring and \(M\) an \(R\)-module let \(R(+) M:=\{\left(\begin{smallmatrix}r &m\\ 0 &r\end{smallmatrix}\right)\mid r\in R,\ m\in M\}\). A description of the \(z\)-ideals of \(R(+) M\) is given too. Moreover there are some results about higher order \(z\)-ideals and it is shown that each SFT-ring is radically \(z\)-covered.
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13C13
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 7495607
zbMATH Keywords: \(z\)-ideal
zbMATH Keywords: higher order \(z\)-ideal
zbMATH Keywords: idealization
zbMATH Keywords: power series
reviewed by: Francois Couchot
MaRDI profile type: Publication
full work available at URL: s13366-020-00553-0
OpenAlex ID: W3120800146
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DOI: 10.1007/S13366-020-00553-0
description: scientific article; zbMATH DE number 7495607