MHB P-Adic Localization in Rational Numbers: A Proof of Local Subring Property

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Hey! :o

Let $p$ be a prime.
We define $$R=\{m/n\in \mathbb{Q}\mid m,n\in \mathbb{Z} \text{ and } p\not\mid n\}$$

I want to show that $R$ is a local subring of $\mathbb{Q}$. To show that, do we have to show that there is a $I\subseteq R$ which satisfies the following conditions?
  1. $I$ is the only maximal right ideal of $R$
  2. $I$ is the only maximal left ideal of $R$
  3. $I$ is an ideal
  4. each element $a\in R-I$ is invertible
(Wondering)
 
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Consider $I = pR$ and use condition 4 to prove $R$ is local.
 

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