Prime ideals of a localization

Theorem: if $\varphi$ is a locatization morphism $A\to A_S$, then $spec \varphi$ is a homeomorphism from $spec(A_S)$ onto the subspace $\{p\in SpecA; p\cap S= \phi$ of $spec A \}$. (Q.Liu, Algebraic geometry, p28 )

Theorem: Let $X$ affine scheme. $X$ is a reduced affine scheme $\Leftrightarrow \sqrt{(0)} =(0) $. (Robin Hartshorn, Algebraic geometry, p82)

We know $\mathbb Z$ is an integral domain, then $\mathbb Z_S$ is also an integral domain. Hence $X=spec \mathbb Z_S$ is an integral affine scheme, then $X=spec \mathbb Z_S$ is reduced and irreducible. i.e. $\sqrt{(0)} =(0)$

Your might translate your answer to the first part, in a more explicit way, as

The bijection maps the primes non-divisors of $n$ to the prime ideals of $\mathbf Z_n$.

For the nilradical question, remember the localisation of an integral domain is an integral domain.

Your proof for the first part looks fine, although you might be a little more explicit about enumerating the prime ideals of $\mathbb{Z}_n$.

For the second part, it might help to remember that the nilradical of a commutative ring with unity is the intersection of all the prime ideals.