TITLE: Localization of a local ring
QUESTION [8 upvotes]: If $A$ is a local ring, $\mathfrak{m}$ is its maximal ideal, then is $A_{\mathfrak{m}}\cong A$ ? That's, in my opinion, because every denominator is invertible. Am I right?

REPLY [10 votes]: Yes you are right. I think the easiest way to formalize this is showing that A satisfies the universal property of the localization.