I got one example on my notes about PID and maximal ideal. I feel it is a strange example as it doesn't make sense to me, and there are no explanations. It says:(adsbygoogle = window.adsbygoogle || []).push({});

For a prime p[itex]\in[/itex][itex]\mathbb{N}[/itex], denote by [itex]\mathbb{Z}_{(p)}[/itex] the subring of [itex]\mathbb{Q}[/itex] given by

[itex]\mathbb{Z}_{(p)}[/itex]={[itex]\frac{m}{n} \in[/itex][itex]\mathbb{Q}[/itex]|p does not divide n}.

Then [itex]\mathbb{Z}_{(p)}[/itex] is a PID, and it has exactly one maximal ideal.

I can't see the reason of this example at all, and I'm not able to imagine what are the ideals like in Z_(p), can anyone please explain to me why it is a PID and only has one maximal ideal? Thanks a lot.

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# Example of PID and its maximal ideal

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