Jacobson radical of $\mathbb{Z}_{(p)}$

Question

Let $\mathbb{Z}_{(p)}$ be the ring of integers localized at a prime ideal $(p)$. What is its Jacobson radical, i.e., the intersection of all its maximal ideals?

Theorem: Let $x\in R$. Then $x$ lies in the Jacobson radical iff $1 - x y$ is a unit for all $y\in R$.

Answer 1

Consider $P=\left\{\dfrac{ap}{b}:a,b\in\mathbb{Z}, b\notin(p)\right\}$ and prove it is a maximal ideal of $\mathbb{Z}_{(p)}$. Are there other maximal ideals?

Answer 2

What is its only maximal ideal? Hint: Try thinking about which elements are not invertible.