Assume that $(R,\mathfrak{m})$ is a commutative local ring of equal characteristic zero. So $R$ contains the field of rationals. The well known $\mathfrak{m}$adic completion of $R$ provides a complete ring $\hat{R}$ whose coefficient field is isomorphic to the residue field of $R$. Do there exists a topological method (completion) for providing a local Noetheiran extension $S$ of $R$ such that $S$ contains the real numbers and also contains $R$ as a dense subring.
