On graded 1-absorbing prime ideals
type: Item
instance of: scholarly article
title: On graded 1-absorbing prime ideals
zbMATH Open document ID: 1472.13001
author: Rashid Abu-Dawwas
author: Suat Koç
author: Eda Yildiz
author: Unsal Tekir
published in: São Paulo Journal of Mathematical Sciences
publication date: 2021-07-28T00:00:00Z
review text: Let \(R\) be a commutative ring graded by a group. A graded 1-absorbing prime ideal of \(R\) is defined by the authors as being a proper graded ideal \(P\) such that for any non-invertible homogeneous elements \(x,y,z\in P\), either \(xy\in P\) or \(z\in P\). Several basic results and equivalent characterizations of such ideals are presented. It is showed that the graded prime ideal of a graded 1-absorbing prime ideal is graded prime. It is proved that if every nonzero proper graded ideal of \(R\) is graded 1-absorbing prime, then every homogeneous element \(x\) of \(R\) is \(\pi\)-regular, i.e., there exist a positive integer \(n\) and \(y\in R\) such that \(x^{2n}y=x^n\). A graded version of the prime avoidance theorem is presented.
reviewed by: Constantin Năstăsescu
Mathematics Subject Classification ID: 13A02
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 7375699
zbMATH Keywords: graded 1-absorbing prime ideal
zbMATH Keywords: graded prime ideal
zbMATH Keywords: graded primary ideal
zbMATH Keywords: graded 2-absorbing ideal
MaRDI profile type: Publication
full work available at URL: s40863-021-00218-3
OpenAlex ID: W3131552368
Wikidata QID: Q107169868
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DOI: 10.1007/S40863-021-00218-3
description: scientific article; zbMATH DE number 7375699