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How does one show that the prime spectrum of a domain is irreducible in the Zariski topology?

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    $0$ is the only minimal prime of $R$. Suppose, $R=V(I)\cup V(J)$. If $I,J$ are nonzero, then the zero ideal is in neither closed set. So, one of them must be zero, in which case, the corresponding closed set is the entire spectrum.2011-02-06
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    A little typo: it's $\text{Spec } R=V(I)\cup V(J)$.2011-02-07
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    The set $\{x\}$ is closed (we say that $x$ is a "closed point") in $\mathrm{Spec} (A) \Leftrightarrow $x is maximal.2014-06-08

3 Answers 3