Fuzzy projective modules and tensor products in fuzzy module categories

Let $R$ be a commutative ring. We write $\mbox{Hom}(\mu_A, \nu_B)$ for the set of all fuzzy $R$-morphisms from $\mu_A$ to $\nu_B$, where $\mu_A$ and $\nu_B$ are two fuzzy $R$-modules. We make $\mbox{Hom}(\mu_A, \nu_B)$ into fuzzy $R$-module by redefining a function $\alpha:\mbox{Hom}(\mu_A, \nu_B)\longrightarrow [0,1]$. We study the properties of the functor $\mbox{Hom}(\mu_A,-):FR\mbox{-Mod}\rightarrow FR\mbox{-Mod}$ and get some unexpected results. In addition, we prove that $\mbox{Hom}(\xi_p,-)$ is exact if and only if $\xi_P$ is a fuzzy projective $R$-module, when $R$ is a commutative semiperfect ring. Finally, we investigate tensor product of two fuzzy $R$-modules and get some related properties. Also, we study the relationships between Hom functor and tensor functor.