Is the zero ideal of a graded ring considered homogeneous?

Question

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.

Answer 1

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.)