This is a follow up question stemming from this prior MSE post.
Given an integral domain $R$, there is a natural inclusion of $R$ into its ring of fractions $\operatorname{Frac}(R)$. Under what conditions is this the only ring homomorphism $R\to \operatorname{Frac}(R)$? In particular, does it suffice for $R$ to be a DVR?
(Note that a necessary condition for this to hold is for $R$ and $\operatorname{Frac}(R)$ to have trivial endomorphism groups.)
I see two examples. When $R = \mathbb{Z}$ this is true because $\mathbb{Z}$ is initial in the category of rings. When $R = \mathbb{Z}_p$, this feels like it ought to be true since $\mathbb{Q}_p$ has no nontrivial ring endomorphisms.