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Intuitionistic Fuzzy Small Submodule with Respect to an Arbitrary Intuitionistic Fuzzy Submodule


Affiliations
1 IKG PT University, Jalandhar, Punjab, India
2 GNDEC, Ludhiana, India
3 Department of Mathematics, D.A.V. College, Jalandhar, Punjab, India
     

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In this paper, we introduce the concept of intuitionistic fuzzy small submodule with respect to an arbitrary intuitionistic fuzzy submodule of an R-module M. We derive the condition when an intuitionistic fuzzy submodule to be a small submodule with respect to another intuitionistic fuzzy submodule with the crisp small submodule of the R-module M.  It is also shown that the sum of two intuitionistic fuzzy small submodules with respect to a fixed intuitionistic fuzzy submodule is again an intuitionistic fuzzy submodule with respect to the same fixed intuitionistic fuzzy submodule.  This result can be extended to an arbitrary sum of intuitionistic fuzzy submodules. Further, we prove that the homomorphic image of an intuitionistic fuzzy small submodule with respect to a fixed intuitionistic fuzzy submodule is again an intuitionistic fuzzy small submodule with respect the homomorphic image of the fixed intuitionistic fuzzy submodule.


Keywords

Intuitionistic Fuzzy Submodule (IFSM), Intuitionistic Fuzzy Small Submodule (IFSSM), B-Small Submodule, Intuitionistic Fuzzy Small Submodule with Respect to another Intuitionistic Fuzzy Submodule.
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  • Intuitionistic Fuzzy Small Submodule with Respect to an Arbitrary Intuitionistic Fuzzy Submodule

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Authors

Gagandeep Kaur
IKG PT University, Jalandhar, Punjab, India
Devinder Singh Pathania
GNDEC, Ludhiana, India
Poonam Kumar Sharma
Department of Mathematics, D.A.V. College, Jalandhar, Punjab, India

Abstract


In this paper, we introduce the concept of intuitionistic fuzzy small submodule with respect to an arbitrary intuitionistic fuzzy submodule of an R-module M. We derive the condition when an intuitionistic fuzzy submodule to be a small submodule with respect to another intuitionistic fuzzy submodule with the crisp small submodule of the R-module M.  It is also shown that the sum of two intuitionistic fuzzy small submodules with respect to a fixed intuitionistic fuzzy submodule is again an intuitionistic fuzzy submodule with respect to the same fixed intuitionistic fuzzy submodule.  This result can be extended to an arbitrary sum of intuitionistic fuzzy submodules. Further, we prove that the homomorphic image of an intuitionistic fuzzy small submodule with respect to a fixed intuitionistic fuzzy submodule is again an intuitionistic fuzzy small submodule with respect the homomorphic image of the fixed intuitionistic fuzzy submodule.


Keywords


Intuitionistic Fuzzy Submodule (IFSM), Intuitionistic Fuzzy Small Submodule (IFSSM), B-Small Submodule, Intuitionistic Fuzzy Small Submodule with Respect to another Intuitionistic Fuzzy Submodule.

References