Let $S$ be a graded ring. Let $\mathfrak p, \mathfrak p' ⊂ S$ be relevant prime ideals.
If $\mathfrak p \cap S_+ = \mathfrak p' \cap S_+$, then $\mathfrak p = \mathfrak p'$.
I don't see how this is always true. Is it not possible that, say, $\mathfrak p$ may have a homogeneous degree $0$ element which is not in $\mathfrak p'$?
Are all degree $0$ elements in $\mathfrak p$ also in $\mathfrak p'$, and vice versa?