Let $R=k[x,y,z]/(xy-z^2)$ and $\mathfrak{m}=(x,y,z)\subset k[x,y,z]$. Set $P\subset \mathfrak m$ be a prime ideal of $k[x,y,z]$ such that $xy-z^2\in P$. I am wondering that if $P\neq \mathfrak m$ then $xy-z^2\notin P^2$ is correct?
I think that it is correct but I can't show it. I suppose that this is true, then the image of $xy-z^2$ is a part of a system of parameters in $k[x,y,z]_P$ so that $R_P$ is a regular local ring (note that $R_{\mathfrak m}$ is not a RLR). Please help clear this up for me!
Since $P$ is prime other than $\mathfrak m$, either $x$ or $y$ is not in $P$. Without loss of generality, we may assume that $x$ is not in $P$. Then $\{x^n\}_{n\ge 0}\cap P=\emptyset$. This yields $(R_x)_P=R_P$. Obviously, $R_x$ is regular so that $R_P$ is a RLR.