Fuzzy chain subrings and their fuzzy direct sum
type: Item
instance of: scholarly article
title: Fuzzy chain subrings and their fuzzy direct sum
zbMATH Open document ID: 0941.13008
DOI: 10.1016/S0165-0114(97)00311-4
published in: Fuzzy Sets and Systems
publication date: 2000-08-13T00:00:00Z
review text: In this paper the author characterizes chain rings in terms of certain fuzzy ideals by using minimal generating sets. He employs the concept of fuzzy direct sum of fuzzy subrings to characterize Artinian principal ideal rings and to find necessary and sufficient conditions under which an Artinian ring is a direct sum of fields. Let \(R\) be a commutative ring with identity. \(R\) is called a chain ring if its ideals form a chain. A fuzzy ideal \(\mu\) of \(R\) is called a fuzzy chain if for all \(x, y\) in \(R\), (1) \(\mu(x)= \mu(y)\) if and only if the ideal \((x) =\) the ideal \((y)\), and (2) \(\mu(x)< \mu(y)\) if and only if \((x)\supset (y)\). The author shows that \(R\) is a chain ring if and only if \(R\) has fuzzy ideal which is a fuzzy chain. The author also shows that the following conditions are equivalent on a fuzzy ideal of an Artinian ring \(R\): (1) \(\mu\) is a fuzzy chain, (2) the level ideal \(\mu_{\mu(x)}\) equals the ideal generated by \(x\) for any \(x\in R\), (3) \(\mu\) is fuzzy local, fuzzy principal, and of maximal chain.
Mathematics Subject Classification ID: 13A99
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 1375041
zbMATH Keywords: chain rings
zbMATH Keywords: fuzzy ideals
zbMATH Keywords: fuzzy chain
author: Yousef Alkhamees
reviewed by: John N. Mordeson
MaRDI profile type: Publication
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full work available at URL: s0165-0114(97)00311-4
OpenAlex ID: W2059116877
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description: scientific article; zbMATH DE number 1375041