Direct products and the contravariant hom-functor

TL;DR

This paper characterizes the zero module by examining when the hom-functor commutes with arbitrary direct products, establishing a fundamental property in module theory within ZFC set theory.

Contribution

It proves that the only module for which the hom-functor commutes with all direct products is the zero module, even under restricted conditions.

Findings

Hom-functor commutes with all direct products only for the zero module.

The result holds in ZFC set theory.

The characterization applies even when all modules in the family are isomorphic to G.

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 $i\in I$.