Let $(R,m)$ be a $d$-dimensional regular local ring with quotient field $K$ and $(S,n)$ be a $d$-dimensional normal local domain birationally dominating $R$ with $l(mS) = d$. In this paper, it is shown that the following three properties hold.
$S$ is dominated by the $m$-adic prime divisor of $R$;
$n^i \cap R = m^i$, for all $i \ge 1$;
$R/m = S/n$.
"Note on Birational Extensions in D-Dimension." Missouri J. Math. Sci. 10 (3) 153 - 158, Fall 1998. https://doi.org/10.35834/1998/1003153