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.

  • 1
    $\begingroup$ Take a peek at $\;pA_P\;$...or at any decent commutative algebra book. $\endgroup$ – Timbuc Dec 6 '14 at 7:58

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

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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.

  • $\begingroup$ Do u mean I is the set of all no unity? And how to get it? $\endgroup$ – annimal Dec 6 '14 at 5:42
  • $\begingroup$ Look at the definition of $A_P$ and make your guess! $\endgroup$ – Andrea Mori Dec 6 '14 at 5:44
  • $\begingroup$ is it $S^{-1}(P)$? $\endgroup$ – annimal Dec 6 '14 at 5:49

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