Exercise 111.27.9. Let $R$ be a graded ring.

Show that $\text{Proj}(R)$ is empty if $R_ n = (0)$ for all $n >> 0$.

Show that $\text{Proj}(R)$ is an irreducible topological space if $R$ is a domain and $R_{+}$ is not zero. (Recall that the empty topological space is not irreducible.)

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