Fuzzy projective modules and tensor products in fuzzy module categories

Abstract

Let R be a commutative ring. We write Hom( A; B) for the set of all fuzzy R-morphisms from A to B, where A and B are two fuzzy R-modules. We make Hom( A; B) into fuzzy R-module by redening a func- tion : Hom( A; B) ! (0; 1). We study the properties of the functor Hom( A; ) : FR-Mod! FR-Mod and get some unexpected results. In ad- dition, we prove that Hom( p; ) is exact if and only if 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.