Let $R$ be a regular local ring of dimension $d$ and let $x_1,x_2,...,x_d$ be a regular system of parameters. Now, for any $y\in R$, the colon ideal $(x_1,x_2,...,x_h):y$ where $h\leq d$ is a prime ideal or the whole ring. I was wondering, if given a prime ideal $P$ in a regular local ring $R$, does there exist a subset of a regular system of parameters, such that $P$ is a colon ideal of the above form?
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Characterization of prime ideals in regular local ringsLet $R$ be a regular local ring of dimension $d$ and let $x_1,x_2,...,x_d$ be a regular system of parameters. Now, for any $y\in R$, the colon ideal $(x_1,x_2,...,x_h):y$ where $h\leq d$ is a prime ideal. I was wondering, if given a prime ideal $P$ in a regular local ring $R$, does there exist a subset of a regular system of parameters, such that $P$ is a colon ideal of the above form?
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