Abstract

In this article, we introduce the notion of fuzzy G-module by defining the group action of G on a fuzzy set of a Z-module M. We establish the cases in which fuzzy submodules also become fuzzy G-submodules. Notions of a fuzzy prime submodule, fuzzy prime G-submodule, fuzzy semi prime submodule, fuzzy G-semi prime submodule, G-invariant fuzzy submodule and G-invariant fuzzy prime submodule of M are introduced and their properties are described. The homomorphic image and pre-image of fuzzy G-submodules, G-invariant fuzzy submodules and G-invariant fuzzy prime submodules of M are also established.

Highlights

  • Introduction and Main ResultsThe concept of fuzzy subset of a non-empty set was introduced by Zadeh [1] who introduced the notion of a fuzzy set as a method of representing uncertainty in real physical world

  • It has been proved that every fuzzy G-module is a fuzzy module but the converse is not true in general

  • It has been proved that intersection and Cartesian product of two fuzzy G-submodules are fuzzy G-submodules

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Summary

Introduction and Main Results

The concept of fuzzy subset of a non-empty set was introduced by Zadeh [1] who introduced the notion of a fuzzy set as a method of representing uncertainty in real physical world. Following this landmark discovery, a number of studies of Fuzzy Modules and their applications have emerged. The notion of group action on fuzzy subset of a ring was defined and studied by Sharma in [4] [5]. We define the group action on fuzzy subset of a module over the ring of integers and introduce the notion of fuzzy G-modules.

Preliminaries Knowledge and Results
Group Action on Fuzzy Modules
Homomorphism of Fuzzy G-Submodules
Conclusion

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