The lattice of \(F^A_R\)-modules
type: Item
instance of: scholarly article
zbMATH Open document ID: 0967.16023
author: Qingde Zhang
author: Tu Yun Chen
author: Xiaozhong Li
publication date: 2001-03-14T00:00:00Z
Mathematics Subject Classification ID: 16Y99
Mathematics Subject Classification ID: 16D99
zbMATH DE Number: 1576304
zbMATH Keywords: fuzzy modules
zbMATH Keywords: fuzzy rings
zbMATH Keywords: complete lattices
zbMATH Keywords: modular lattices
MaRDI profile type: Publication
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title: The lattice of \(F^A_R\)-modules
published in: Journal of Liaoning Normal University. Natural Science Edition
review text: The authors elaborate on the notion of a fuzzy module introduced by \textit{C. V. Negoita} and \textit{D. A. Ralescu} in their pioneering 1975 book [on the Applications of Fuzzy Sets to System Analysis (Birkhäuser-Verlag, Basel, 1975; Zbl 0326.94002)]. The connection between a fuzzy module and a fuzzy ring has been established by J. Zhao in introducing the so-called \(F^A_R(M)\)-modules, i.e. fuzzy submodule of a left \(R\)-module \(M\) over a fuzzy subgring \(A\) of a ring \(R\).NEWLINENEWLINENEWLINEHere, the authors investigate these \(F^A_R\)-modules from a lattice theoretic point of view. In particular, the following properties have been proved: -- \(F^A_R(M)\) forms a complete lattice w.r.t. Zadeh's sharp inclusion relation; -- the sum or sup-min composition of two \(F^A_R\)-modules of \(M\) also constitutes an \(F^A_R\)-module; -- \(F^A_R(M)\) constitutes a modular lattice; -- the direct and inverse image of an \(F^A_R\)-module under a homomorphism constitute again \(F^A_R\)-modules. -- Finally it is stated, however without proof, that the lattice \(F^A_R(M)\) is not distributive.
reviewed by: E. E. Kerre
description: scientific article