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Suppose $A$ is a Dedekind domain with fraction field $K$ and $L/K$ is Galois, let $B$ be the integral closure of $A$ in $L$.

Let $P$ be a prime ideal in $A$ and let $P_1,...,P_n$ be prime ideals lying over $P$ in $B$. Then, $P=(P_1\cap ... \cap P_n)\cap A$. I was wondering if it's true that $(P_1^i \cap ... \cap P_n ^i)\cap A$ is primary?

PS: I was posting with username BMI earlier. For some reason, every time I posted this question, I got an error "this question could not be submitted because it does not meet our quality standards". I copy pasted it from a new user id and it posted without a hassle.

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up vote 0 down vote accepted

Let's denote the ideal $(P_1^{i_1} \cap ... \cap P_n^{i_n}) \cap A$ by $I.$ Let $i = \max\{i_1,..,i_n\}.$ As $P^i \subset P_k^i$ for all $k,$ $P^i \subset I.$ It follows $I | P^i,$ so $I =P^j$ for some $j\le i.$ Hence, $I$ is P-primary.

Note that the assumption that $L/K$ is galois was unnecessary as was the assumption that $i_1 = i_2 =... = i_n.$

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Thanks, jspecter. I wonder if the conclusion will still hold if we don't assume $A$ is a Dedekind domain. I don't think $I$ would still be $P^j$ in general if we relax that assumption. – NewB Oct 18 '11 at 1:31

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