Is the zero ideal of a graded ring considered homogeneous? I'm working through Vakil's notes, and he defines a $\Bbb{Z}^{\ge0}$-graded ring
$$S_{\bullet}=\oplus_{n\ge0} S_n$$
and $X:=\operatorname{Proj}S_{\bullet}$ to be the collection of homogeneous prime ideals of $S_{\bullet}$ not containing $S_+:=\oplus_{n>0}S_n$.
My question is:

If $S_{\bullet}$ is a domain, is the zero ideal contained in $X$? 

Of course $(0)$ is prime and doesn't contain $S_+$, so the question is if it is homogeneous. An ideal $I\subseteq S_{\bullet}$ to be homogenous if the two following equivalent conditions hold:
$(1)$ $I$ is generated by homogeneous elements.
$(2)$ $I$ contains all homogeneous components of each of its elements.
It seems that $(2)$ is trivially true for the zero ideal, but condition $(1)$ confuses me here. Is $0$ considered a homogenous element? Wikipedia says that $0$ is homogeneous of degree zero but this seems to contradict the condition $S_nS_m\subseteq S_{m+n}$. Also, if $0$ is homogeneous of degree zero, then it can't be contained in $S_+$, even though this is supposed to be an ideal.
Can anybody clear up my confusion? There's nothing too complicated going on here I just seem to be confused about some definitions.
 A: Yes, the zero ideal is homogeneous.  The element $0\in S_\bullet$ is homogeneous, but its degree is not uniquely defined--it is "homogeneous of degree $n$" simultaneously for all $n$.  After all, it is an element of $S_n$ for all $n$.  Of course, the degree of any nonzero homogeneous element is unique, since $S_n\cap S_m=\{0\}$ if $m\neq n$.
The context in which Wikipedia says $0$ is homogeneous of degree zero is in relation to the fact that $S_0$ is a subring of $S_\bullet$: in order to be a subring, it must contain $0$.  This does not imply $0$ can't be homogeneous of other degrees as well though!
Note moreover that you don't even need to say $0$ is homogeneous in order for (1) to hold, since the ideal $\{0\}$ is generated by the empty set.  Every element of the empty set is homogeneous.
(In fact, it would be not entirely unreasonable to define a homogeneous element of $S_\bullet$ to be a nonzero element of some $S_n$.  The idea behind this definition is that a homogeneous element is an element which has exactly one nonzero component.  So you would not consider $0$ to be homogeneous, similar to how $1$ is not considered to be a prime number.  I can imagine there might be circumstances where this definition is the more natural one.  But as far as I know, the standard definition is that $0$ is homogeneous.)
