How should we define the function $I(\cdot)$ for ${\rm Proj} S_\bullet$(the homogeneous prime ideals not containing $S_+$) for a $\mathbb{Z}^{\ge 0}$-graded ring $S_\bullet=S_0\oplus S_+$?
I know the functions $V(\cdot)$ and $I(\cdot)$ for affine schemes. I want the projective version of them. I know the projective version of $V(\cdot)$, i.e. $V(T):=\{p| p\supset T\}\subset {\rm Proj} S_\bullet$ for $T\subset S_+.$
If we follow the affine case, we might define $I(Z):=\cap_{p\in Z}p$ for $Z\subset {\rm Proj} S_\bullet$.
However this definition does not satisfy $I(Z)\subset S_+$ because if $S_0:=\mathbb{Z}, S_\bullet:=\mathbb{Z}[x]$ and $Z=\{ (2)\}$ then $I((2))=(2)=(2\mathbb{Z})[x]\supset 2\mathbb{Z} \not\subset S_+.$
$\bullet {\bf EDIT}$ (added just after my first comment to the first answer):
How about the following? $I(Z):=\langle (\cap_{p\in Z}p)\cap \cup_{i>0}S_i\rangle$ Here $\cup_{i>0}S_i$ means the homogeneous elements of positive degree, and the bracket means the ideal generated by the ingredients.
How do people define $I(\cdot)$ usually?