Reduced, irreducible and integral schemes

Question

I am studying properties of schemes and I have the following problem.

Let $R$ be a ring, $S=Spec(R)$, $n\in\mathbb{N}$ an integer. Show that the following are equivalent:

1. $S$ is reduced (resp irreducible, resp integral)
2. $\mathbb{A}^n_R$ is reduced (resp irreducible, resp integral)
3. $\mathbb{P}^n_R$ is reduced (resp irreducible, resp integral)

I know that a scheme $S=Spec(R)$ is reduced if and only if nil$(R)=0$; it is irreducible if and only if nil$(R)$ is prime; and it is integral if and only if it is both reduced and irreducible.

If someone can help me, it would be very appreciated.