I'm working on an exercise in which I have to show that localising and completing are exact functors. More precisely I have a Dedekind domain $R$ and a prime ideal $\mathfrak{p}$ and I have to show that $M\mapsto R_\mathfrak{p} \otimes_R M$ and $M\mapsto \hat{R_\mathfrak{p}} \otimes_R M$ ($R_\mathfrak p$ the localisation at $\mathfrak p$, $\hat{R_\mathfrak p}$ the completion) preserve exact sequences.
My question in this context is the following: is it true that the localisation $R_\mathfrak{p}$ is contained in the completion $\hat{R_\mathfrak{p}}$ (using a suitable embedding)? I understand that this is true for the ring $\mathbb{Z}$, the localisation $\mathbb{Z}_{(p)}$ ($p$ a prime number) and the ring of $p$-adic integers, so in other words we have $\mathbb{Z}\subset \mathbb{Z}_{(p)} \subset \mathbb{Z}_p$. Is this correct in the case of an arbitrary Dedekind domain as well? Is it maybe also true in a more general setting, say if $R$ is "just" an integral domain?
I'm looking forward to your answers.