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Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x_1,x_2,...,x_n$ s.t. $m=(x_1,x_2,...,x_n)R$. Further, the ideals $(x_{i_1},...,x_{i_j})$ with $i_1,...,i_j\in {1,...,n}$, are prime.

Can we make similar statements about any other kind of prime ideals in a regular local ring $R$. R$? Specifically, do any other prime ideals satisfy the condition: if the ideal is minimally generated by a certain set of generators, then every subset of the generators defines a prime ideal. ? One example in light of the first paragraph, are the prime ideals generated by a subset of the regular sequence that generates the maximal ideal. Also, when does a regular sequence define a prime ideal in a regular local ring$R$R$, and when does a maximal regular sequence define a maximal ideal.?

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Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x_1,x_2,...,x_n$ s.t. $m=(x_1,x_2,...,x_n)R$. Further, the ideals $(x_{i_1},...,x_{i_j})$ with $i_1,...,i_j\in {1,...,n}$, are prime.

Can we make similar statements about any other kind of prime ideals in a regular local ring $R$. Specifically, do any other prime ideals satisfy the condition: if the ideal is minimally generated by a certain set of generators, then every subset of the generators defines a prime ideal. One example in light of the first paragraph, are the prime ideals generated by a subset of the regular sequence that generates the maximal ideal.

Also, when does a regular sequence define a prime ideal in a regular local ring $R$ and when does a maximal regular sequence define a maximal ideal.

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# prime ideals in regular local rings

Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x_1,x_2,...,x_n$ s.t. $m=(x_1,x_2,...,x_n)R$. Further, the ideals $(x_{i_1},...,x_{i_j})$ with $i_1,...,i_j\in {1,...,n}$, are prime.

Can we make similar statements about any other kind of prime ideals in $R$. Specifically, do any other prime ideals satisfy the condition: if the ideal is minimally generated by a certain set of generators, then every subset of the generators defines a prime ideal. One example in light of the first paragraph, are the prime ideals generated by a subset of the regular sequence that generates the maximal ideal.

Also, when does a regular sequence define a prime ideal in $R$ and when does a maximal regular sequence define a maximal ideal.