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

Question

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.

Answer 1

Book: "Steps in Commutative Algebra" by "R.Y. Sharp"

Answer 2

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.