# $A$ is a commutative ring, $P$ is a prime ideal. Prove $A_P$ is local ring

Question: Suppose $A$ is a commutative ring, $P$ is a prime ideal. Prove $A_P$ is local ring.

I have no idea how to construct the unique maximal ideal.

• Take a peek at $\;pA_P\;$...or at any decent commutative algebra book. – Timbuc Dec 6 '14 at 7:58

HINT: Let $R$ be a commutative ring with $1$, and let $I\subset R$ an ideal. If all the elements in $R\setminus I$ are invertible in $R$ then $R$ is local and $I$ is its unique maximal ideal.
• Look at the definition of $A_P$ and make your guess! – Andrea Mori Dec 6 '14 at 5:44
• is it $S^{-1}(P)$? – annimal Dec 6 '14 at 5:49