On fuzzy fractions formation

Let R be a commutative ring, S a multiplicatively closed subset of R and let $S^{-1}R$ be the ring of fractions over S. For a typical fuzzy submodule η of the R-module M, the fuzzy fractions $S^{-1}\eta$ of μ is a fuzzy submodule of the $S^{-1}R$-module (of fractions) $S^{-1}M$. In this note, it is shown that if $\mu \subseteq \lambda$ are two fuzzy submodules of M, both having sup property, then there exist a (fuzzy sense) isomorphism between $S^{-1}(\lambda /\mu )$ and $S^{-1}\lambda /S^{-1}\mu$. Then, it is seen that local global principle holds true for fuzzy submodules. Furthermore, after providing some auxiliary results, it is proved that under some mild conditions, fuzzy fractions formation commutes with fuzzy residual quotient.