Let $R := k[X_1,\dots,X_n]/I$ be a quotient of a polynomial ring by some prime ideal, $\mathfrak{p} \subset \mathfrak{m}$ two ideals of $R$, where $\mathfrak{p}$ - prime ideal and $\mathfrak{m}$ - maximal ideal containing it. What is the easiest proof that if $R_{\mathfrak{m}}$ (localization of $R$ by $\mathfrak{m}$) is a regular local ring, then $R_{\mathfrak{p}}$ is also a regular local ring.
There is well-known result that if $A$ is regular local ring then so is any of its localizations by prime ideals. There are proofs of this in many references, but it is very non-elementary. So can someone indicate an easier proofs or give some references for the special case I given above.
Thank you!