Fuzzy semistar operations on integral domains
type: Item
instance of: scholarly article
title: Fuzzy semistar operations on integral domains
zbMATH Open document ID: 1263.13025
author: Olivier A. Heubo-Kwegna
published in: Fuzzy Sets and Systems
publication date: 2012-12-04T00:00:00Z
review text: The author introduces the notion of a fuzzy semistar operation on an integral domain and relates it to the existing notion of a fuzzy star operation. He shows that the set of all fuzzy semistar operations on the integral domain is a complete lattice. He also characterizes when an overring of a domain is a Prüfer domain and when an overring is a flat module over the domain in terms of a fuzzy semistar operation. Let \(R\) be an integral domain. Let \(F_z(R)\) denote the set of all fractionary fuzzy ideals of \(R\) and \(\overline F_z(R)\) the set of all fuzzy \(R\)-submodules of \(K\), where \(K\) is the quotient field of \(R\). A fuzzy semistar operation on \(R\) is a function \(*:\overline F_z(R)\to\overline F_z(R)\) such that the following properties hold for all \(\alpha,\beta\in\overline F_z(R)\) and \(0\neq d\in K\): \((*_1)\) \((d_1\circ\beta)^*=d_1\circ\beta^*\); \((*_2)\) \(\alpha\subseteq\beta\Rightarrow \alpha^*\subseteq\beta^*\); \((*_3)\) \(\beta\subseteq\beta^*\) and \(\beta^{**}=\beta^*\). In addition to the results stated above, the author examines the extensions of fuzzy star operations to fuzzy semistar operations and when a fuzzy semistar restricts to a fuzzy star operation. The author also fuzzifies some classical results of multiplicative ideal theory with the use of a fuzzy semistar operation. He shows that an overring \(T\) of \(R\) is a flat \(R\)-module if and only if the fuzzy star operation \(*_T\) is stable.
reviewed by: John N. Mordeson
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 03E72
zbMATH DE Number: 6112280
zbMATH Keywords: fuzzy submodule
zbMATH Keywords: fractionary fuzzy ideal
zbMATH Keywords: fuzzy star operation
zbMATH Keywords: fuzzy semistar operation
zbMATH Keywords: flat-module
zbMATH Keywords: Prüfer domain
MaRDI profile type: Publication
full work available at URL: j.fss.2012.03.007
OpenAlex ID: W2063815614
DOI: 10.1016/J.FSS.2012.03.007
description: scientific article; zbMATH DE number 6112280