Spectrum of the ring of formal power series over integers Let $\mathbb{Z}[[X]]$ be the formal power series ring over $\mathbb{Z}$.  
I want to understand the set of prime idelas $\rm{Spec}(\mathbb{Z}[[X]])$, maximal ideals $\rm{Spm}(\mathbb{Z}[[X]])$ and Jacobson radical $J(\mathbb{Z}[[X]])$.  
Can you write these sets explicitly $??$
 A: Maximal ideals are of the form $(p,X)$ for a prime $p$ (see Wiki) and the other primes are just $(p)$, $(X)$ and $0$, for $p$ prime. The Jacobson radical is the intersection of the maximal ideals, so it equals $(X)$. Alternatively there is a proof at Jacobson radical of formal power series over an integral domain.
More geometrically and to prove it, you can just think about the fibers of the map :$$\text{Spec}\, \mathbb Z[[X]] \to \text{Spec}\,\mathbb Z$$


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*In the generic fiber are primes such that $\mathfrak p \cap \mathbb Z = 0$, which correspond to primes in $\mathbb Z[[X]]_{(0)} = \mathbb Q[[X]]$.

*In the fiber over $(p)$ are primes such that $\mathfrak p \cap \mathbb Z = (p)$, which correspond to primes in $\mathbb Z_{(p)}[[X]]/(p) = \mathbb F_p[[X]]$.
Any field $k$ has formal power series $k[[X]]$ a DVR with just two prime ideals: $(0) \subset (X)$. This shows that the prime ideals are all of the form $(X,p), (X), (p), 0$ for a prime $p$.
This is similar to how you may think about $\text{Spec}\,\mathbb Z[X]$ (see Spectrum of $\mathbb{Z}[x]$) but even a little easier.
