Simple Calculation on Local Rings.

Question

Let $p$ be prime and $\mathbb{Z}_{(p)}$ be the local ring. I already know, that \begin{align} \mathbb{Z}_{(p)}/p\mathbb{Z}_{(p)} \cong \mathbb{Z}/p\mathbb{Z}. \end{align}

1. What ist the explicit map?

2. There exists some $t \in \mathbb{Z}$ with $p|t$ such that $\mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)} \cong \mathbb{Z}/t\mathbb{Z}$.

Answer 1

Hint for #1: Obviously $a$ maps to $a+p\Bbb Z$ for $a\in\Bbb Z$. Suppose $a/b$ maps to $c+p\Bbb Z$. Then what can you say about $bc$ mod $p$? What does this tell you about $c$? (This is why it's important $p\nmid b$.)

Hint for #2: Prove $q\,\Bbb Z_{(p)}=\Bbb Z_{(p)}=q^{-1}\Bbb Z_{(p)}$ for any prime $q\ne p$. Then argue $t\Bbb Z_{(p)}=p^{v_p(t)}\Bbb Z_{(p)}$.