# Homogeneous prime ideals and grade zero elements

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.

• If P is a prime ideal and $I\cap J\subseteq P$ then what can conclude? Furthermore, what if $J$ is not contained in P? – user26857 Dec 7 '18 at 16:32

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 $$\mathfrak{p} \in V(I) \Leftrightarrow \mathfrak{p} \in V(I \cap R_+)$$.