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The lattice of \(F^A_R\)-modules. (English) Zbl 0967.16023

The authors elaborate on the notion of a fuzzy module introduced by C. V. Negoita and 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 \(R\).
Here, 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.
Reviewer: E.Kerre (Gent)

MSC:

16Y99 Generalizations
16D99 Modules, bimodules and ideals in associative algebras

Citations:

Zbl 0326.94002
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