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 $\mathfrak{q}$?



integral closure of parameter ideals
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".


