Zero set of a homogeneous element of degree $0$, or how $D_+(2)\subset \text{Proj}(\mathbb{Z}[x])$ looks like. Let $S=\bigoplus_{n=0}^\infty S_n$ be a graded ring. We denote $S_+=\bigoplus_{n>0}^\infty S_n$. As usual we define $\text{Proj}(S)$ to be the set of homogeneous,
prime ideals $\mathfrak p$ of $S$ such that
$S_{+} \not \subset \mathfrak p$. The usual definition goes as follows: for $f \in S$ homogeneous of degree $> 0$ we define
$$
D_{+}(f) = \{ \mathfrak p \in \text{Proj}(S) \mid f \not\in \mathfrak p \}.
$$
In Vakil's notes is stated that the definition makes sense also for $f$ of degree zero, but in that case the resulting $D_{+}(f)$ may not be an affine scheme.
I am trying to see one of such examples. My problem is that the only projective schemes I am familiar with come from homogeneous polynomials with coefficients in a field. In that case if $f$ has degree $0$, the set $D_{+}(f)$ is empty. I tried to think about $D_+(2)\subset \text{Proj}(\mathbb{Z}[x])$ but I have a hard time trying to figure it out. Can you help me?  Do you have easier examples of this phenomenon?
 A: The prime ideals of $\mathbb{Z}[x]$ come in one of 3 types: $(p)$, where $p \in \mathbb{Z}$ is prime; $(f(x))$, where $f(x) \in \mathbb{Z}[x]$ is irreducible; $(p, f(x))$, where $p \in \mathbb{Z}$ is prime and $f(x) \in \mathbb{Z}[x]$ is irreducible mod $p$. Then, the distinguished open $D(2)$ in $\textrm{Spec}(\mathbb{Z}[x] )$ consists of the primes that do not contain $2$, i.e. 
$$
D(2) = \{ (p) \colon p \not= 2 \} \cup \{ (f(x)) \colon \textrm{ $f$ irreducible } \}\cup \{ (p, f(x)) \colon p \not= 2 \textrm{ and $f$ irreducible} \}.
$$
Now for the Proj case: $D_+(2)$ consists of the homogeneous prime ideals that do not contain $2$, so we can discard the primes from the above list which are not homogeneous. That is,
$$
D_+(2) = \{ (p) \colon p \not= 2 \} \cup \{ (f(x)) \colon \textrm{ $f$ irreducible and homogeneous } \} \cup \{ (p,f(x)) \colon p \not= 2 \textrm{ and $f$ homogeneous and irreducible mod $p$ } \}.
$$
As Ayman Hourieh points out, the ideals of the form $(p,f(x))$ are still homogeneous ideals if they are generated by homogeneous elements. 
As Martin Brandenburg pointed out, we have not discussed the sheaf of rings on $D_+(2)$ or on $\textrm{Proj}(\mathbb{Z}[x] )$, but hopefully seeing the underlying topological space of the distinguished open $D_+(2)$ will be helpful!
