Let $R = \bigoplus_{d \in \mathbb{N}_0}R^{(d)} $ be a graded ring and for homogeneous ideals $I$, let $V_{proj\;R}(I) = \{p \supseteq I \mid p \text{ is a homogeneous prime ideal and } p \nsupseteq R_+\}$ where $R_+ = \bigoplus_{d \in \mathbb{N}} R^{(d)}$ is the irrelevant ideal. I want to show that $V_{proj\;R}(I) = V_{proj\;R}(I \cap R_+)$.
I have already shown that $I = \bigoplus_{d \in \mathbb{N}_0} I \cap R^{(d)}$ holds, but I currently fail to see why the grade zero elements are uniquely determined by the elements of higher grade and am therefore looking for hints on this question.
As user26857 hinted at, if $I \cap R_+ \subset \mathfrak{p}$, then also $I \cdot R_+ \subset \mathfrak{p}$. Hence either $I \subset \mathfrak{p}$ or $R_+ \subset \mathfrak{p}$, because $\mathfrak{p}$ is prime. The latter is excluded for all primes in $V_{\text{proj }R}(I \cap R_+)$, so we see that $\mathfrak{p} \in V(I) \Leftrightarrow \mathfrak{p} \in V(I \cap R_+)$.