Multiplication modules and tensor product

All rings are commutative with identity and all modules are unital. The tensor product of projective (resp. flat, multiplication) modules is a projective (resp. flat, multiplication) module but not conversely. In this paper we give some conditions under which the converse is true. We also give necessary and sufficient conditions for the tensor product of faithful multiplication Dedekind (resp. Prüfer, finitely cogenerated, uniform) modules to be a faithful multiplication Dedekind (resp. Prüfer, finitely cogenerated, uniform) module. Necessary and sufficient conditions for the tensor product of pure (resp. invertible, large, small, join principal) submodules of multiplication modules to be a pure (resp. invertible, large, small, join principal) submodule are also considered.