Question. Let $R$ be a ring for which the unique homomorphism of rings $ℤ \to R$ is an epimorphism. Is every additive map between $R$-modules already $R$-linear?
Context. Part of Exercise 1.5.xi in Emily Riehl’s Category Theory in Context asks if the forgetful functor $\mathsf{Mod}_R \to \mathsf{Ab}$ is full. In other words, if every additive map between $R$-modules is already $R$-linear. The answer depends on $R$. For example, this property holds
if $ℤ \to R$ is surjective. This means that $R ≅ ℤ/n$ for some $n ≥ 0$.
if $R = ℚ$. More generally if $R$ is a localizations of $ℤ$.
Both types of example rely on the fact that $ℤ \to R$ is “surjective enough”. So I’m wondering if this property holds more generally whenever $ℤ \to R$ is an epimorphism.
