Abstract

In this paper, we introduce the notion of $\mathfrak{F}$-closure of intuitionistic fuzzy submodules of a module $M$. Our attempt is to investigate various characteristics of such an $\mathfrak{F}$-closure. If $\mathfrak{F}$ is a non-empty set of intuitionistic fuzzy ideals of a commutative ring $R$ and $A$ is an intuitionistic fuzzy submodule of $M$, then the $\mathfrak{F}$-closure of $A$ is denoted by $Cl_{\mathfrak{F}}^{M}(A)$. If $\mathfrak{F}$ is weak closed under intersection, then (1) $\mathfrak{F}$-closure of $A$ exhibits the submodule character, and (2) the intersection of $\mathfrak{F}$-closure of two intuitionistic fuzzy submodules equals the $\mathfrak{F}$-closure of intersection of the intuitionistic fuzzy submodules. If $\mathfrak{F}$ is weak closed under intersection, then the submodule property of $\mathfrak{F}$-closure implies that $\mathfrak{F}$ is closed. Moreover, if $\mathfrak{F}$ is inductive, then $\mathfrak{F}$ is a topological filter if and only if $Cl_{\mathfrak{F}}^{M}(A)$ is an intuitionistic fuzzy submodule for any intuitionistic fuzzy submodule $A$ of $M$.

Keywords and Phrases

Intuitionistic fuzzy ideals(submodules), $\mathfrak{F}$-closure, $\mathfrak{F}$-torsion, $\mathfrak{F}$-closed, topological filter.

A.M.S. subject classification

08A72, 03E72, 03F55, 16D10, 13C12.

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