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 [۰,۱]. 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.

Keywords:

Fuzzy set , Hom functor , Fuzzy projective R-module , Fuzzy R-module , Tensor product , functor