Non-zero prime ideal of ring of integers is invertible

Question

I am looking at a lemma in my class note under "The Field of Fractions of Dedekind Domain" as follow:

LEMMA: Assume that $R$ is an integrally closed noetherian domain of Krull dimension 1. Then each non-zero prime ideal of $R$ is invertible.

I would like to better appreciate this lemma by doing it the dumb way: Giving myself with an example, and looks like $\mathbb Z$ is the simplest example:

(a) I read somewhere that $\mathbb Z$ is notherian domain of dimension 1.
(b) $\mathbb Z$ is integrally closed in $\mathbb Q$, because $\mathbb Q$ is integral closure of $\mathbb Z$. (I am not sure if I saying it correctly here.).
(c) The simplest non-zero prime ideal of $\mathbb Z$ is $(2) = 2 \mathbb Z$.
(d) Having said all the above, how should I express mathematicaly that the prime ideal $2\mathbb Z$ is invertible?

Thank you all very much for your time and effort.

Answer 1

Having said all the above, how should I express mathematicaly that the prime ideal $2ℤ$ is invertible?

As the wiki page mentions you could just compute $(\Bbb Z:(2))=\{x\in \Bbb Q\mid (2)x\subseteq \Bbb Z\}=\frac12\Bbb Z\subseteq \Bbb Q$, and see that $(2)(\Bbb Z:(2))=\Bbb Z$.