Let $R$ be a commutative Noetherian ring with unit. Is it true that two prime ideals $p$ and $q$ are in the same connected component of $\text{Spec} R$ iff there exists a series of minimal primes ideals $p_1,\dots,p_n$ with $p\in \overline{p}_1$, $q\in\overline{p}_n$, $\overline{p}_i\cap\overline{p}_{i+1}\neq \emptyset$ for $1\leq i\leq n-1$. (Here $\overline{p}$ denotes the closure of $p$ in $\text{Spec} R$, that is, the set of primes $q$ with $p\subseteq q$.)
Prime ideals in the same connected component of Spec
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algebraic-geometry
commutative-algebra
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0@navigetor23 Thanks for the reference, i will look at the hint. – 2012-11-07