Meaning of ring notation $\Bbb Z_{(p)}$ in commutative algebra [localization]

Question

I am doing some exercises on commutative algebra and came along the following expressions, which were not elaborated on. Is someone familiar with them?

The first is for $p$ a prime number $\mathbb{Z}_{(p)}$, where the exercise is to determine all ideals. Do you think this is just $\mathbb{Z}/p\mathbb{Z}$?

The second one is for $P$ a prime ideal the expression $P_P$, where $P_P$ lies in $R_P$ for $R$ a ring.

I never came across this notation before so I have to ask here.

Answer 1

The ring $\mathbb{Z}_{(p)}$ is defined as $\{ \frac{a}{b}\in \mathbb{Q}\mid p\nmid b\}$.

In general if $R$ is an integral ring and $P$ a prime ideal, $R_P:= \{ \frac{a}{b}\in\operatorname{Frac}(R)\mid b\not\in P\}$.

Answer 2

1. $\mathbb Z_{(p)}$ is the localization of $\mathbb Z$ at the prime ideal $p\mathbb Z$.

2. $P_P$ stands for the maximal ideal of $R_P$. Other (better?) notation: $PR_P$.