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Let $(R, \mathfrak{m})$ be an excellent domain of dimension $d$. Let $\mathfrak{q} = (x_1,...,x_d)$ be a parameter ideal of $R$.

Question: Is it true that $(x_1,...,x_{d-1}):x_d$ is contained in the integral closure of $\mathfrak{q}$?

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A result proved by Ratliff shows that in any locally formally equidimensional noetherian ring $(x_1,...,x_{d-1}):x_d$ is contained in the integral closure of $(x_1,...,x_{d-1})$. See Theorem 1.6.6 from the book of Huneke and Swanson "Integral Closure of Ideals, Rings, and Modules".

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Thanks you very much. –  Pham Hung Quy Oct 3 '13 at 7:47

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