Direct products and the contravariant hom-functor

Abstract

We prove in ZFC that if $G$ is a (right) $R$-module such that the groups $\Hom_R(\prod_{i\in I}G_i,G)$ and $\prod_{i\in I}\Hom_R(G_i,G)$ are naturally isomorphic for all families of $R$-modules $(G_i{)}_{i\in I}$ then G=0. The result is valid even we restrict to families such that $G_i\cong G$ for all $i\in I$.