3 added 77 characters in body

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 completion local Noetheiran extension $S$ of $R$ such that $S$ contains the real numbers ?and also contains $R$ as a dense subring.

2 deleted 4 characters in body

Assume that $(R,\mathfrak{m})$ is a commutative local ring of 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 coefficient residue field of $R$. Do there exists a topological method for providing a completion $S$ of $R$ such that $S$ contains the real numbers?

1

# Completion of commutative rings.

Assume that $(R,\mathfrak{m})$ is a commutative local ring of 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 coefficient field of $R$. Do there exists a topological method for providing a completion $S$ of $R$ such that $S$ contains the real numbers?