One dimensional local domains and radical formula
type: Item
instance of: scholarly article
title: One dimensional local domains and radical formula
zbMATH Open document ID: 1329.13019
author: Anand Parkash
published in: Beiträge zur Algebra und Geometrie
publication date: 2015-10-16T00:00:00Z
review text: Let \(R\) be a commutative ring with identity and \(M\) an \(R\)-module. Let \(P\) and \(N\) be proper submodules of \(M\). We say that \(P\) is a prime submodule whenever \(re\in P\) for \(r\in R\) and \(e\in M\), we have \(r\in (P:_RM)\) or \(e\in P\). The intersection of all prime submodules of \(M\) containing \(N\) is denoted by \(\text{rad}(N)\). Also, the envelope of \(N\) is denoted by \(E(N)\) and defined to be the set \[ \{x\in M|\, x=ay, a^ny\in N\, \text{for some}\, a\in R, y\in M\, \text{and}\, n\in\mathbb{N}\}. \] We say that \(M\) satisfies the radical formula if for every submodule \(L\) of \(M\), \(\text{rad}(L)=\langle E(L) \rangle\); and we say that \(R\) satisfies the radical formula if every \(R\)-module satisfies the radical formula. In this paper, the author shows that a local integral domain \((R, \mathfrak m)\) of dimension one satisfies the radical formula if and only if for all integers \(n\geq2\), and for every \(a_1, \dots, a_n\in R\) with \(a_i\not \in Ra_j\) for all \(i\neq j\), \[ (Ra_1:Ra_2+\dots+Ra_n)=\mathfrak m((Ra_1:Ra_2+\dots+Ra_n). \]
reviewed by: Monireh Sedghi
Mathematics Subject Classification ID: 13C13
Mathematics Subject Classification ID: 13C99
zbMATH DE Number: 6495596
zbMATH Keywords: prime submodules
zbMATH Keywords: radical formula
zbMATH Keywords: local rings
zbMATH Keywords: integral domain
MaRDI profile type: Publication
full work available at URL: s13366-014-0189-3
OpenAlex ID: W1965859675
cites work: On the prime radical of a module over a commutative ring
cites work: On commutative Noetherian rings which satisfy the radical formula
cites work: One dimensional domains which satisfy the radical formula are Dedekind domains
cites work: On commutative noetherian rings which satisfy the generalized radical formula
cites work: On radicals of submodules
cites work: Arithmetical rings satisfy the radical formula
DOI: 10.1007/S13366-014-0189-3
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description: scientific article; zbMATH DE number 6495596