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Let $\mathbb{Z}_{(p)} = \left\{\frac{a}{b}\in\mathbb{Q}:p\nmid b\right\}$

How is it posible to show that $\mathbb{Z}_{(p)}/p\mathbb{Z}_{(p)}$ is ismorphic to $\mathbb{Z}/p\mathbb{Z}$.

I want to show there is sujective homomorphism $\mathbb{Z}_{(p)}/p\mathbb{Z}_{(p)}$ to $\mathbb{Z}/p\mathbb{Z}$. But I'm unable to get the correct map.

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The other way round is easier: there is are obvious maps $\mathbb{Z} \rightarrow \mathbb{Z}_{(p)} \rightarrow \mathbb{Z}_{(p)} / p \mathbb{Z}_{(p)}$. One checks that the composition is surjective with kernel $(p)$; this gets what you want.