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Let $A$ be a ring, let $\mathfrak{p}$ be a prime ideal of $A$ and let $\mathfrak{m}$ be a maximal ideal of $A$ containing $\mathfrak{p}$. Show that $ A_{\mathfrak{p}} = \left( A_{\mathfrak{m}} \right)_{\mathfrak{p}A_{\mathfrak{m}}} $. Conclude that the local ring at a non-closed point (of a scheme) is the localization of a local ring at a closed point.