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

[mathmari]

Hey!

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)

 

[Euge]

Consider $I = pR$ and use condition 4 to prove $R$ is local.