Let $R$ be a Noetherian local domain which is not a UFD and let $P$ be a height one prime ideal of $R.$ Can we find an element $x\in P$ such that $P$ is the only minimal prime ideal containing $x$?
Height one prime ideal of arithmetical rank greater than 1
4
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commutative-algebra
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0If we use Krull's height Theorem, then we can say that $P$ is minimal over an element $y\in R$. Since $R$ is Noetherian, we know that there are only finitely many minimal primes $P_i$ over (y). We may find a new element $x$ that avoids all these $P_i$, but there might be other primes that might be minimal over this new element $x$. So now i feel that the above statement might be false, but i have no counterexample. – 2012-06-21
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0Related to [Regular local ring and a prime ideal generated by a regular sequence up to radical](http://math.stackexchange.com/questions/163700/regular-local-ring-and-a-prime-ideal-generated-by-a-regular-sequence-up-to-radic). – 2013-04-28