# primary ideal of regular local ring

Let $(R,\mathfrak{m})$ be a regular local ring of dimension $d$. Let $P$ be a prime ideal of height $d-1$. I want to know if $P^2$ is always a $P$ primary ideal ie if $P/P^2$ is torsion free as $R/P$ module. Thanks.

Let me explain few of my observations. Assume that $R$ is a complete local ring. Then by Cohen's structure theorem $R$ is a power series ring over a DVR. In eqicharacteristic case it is actually a power series over a field. So can we prove the statement in the following special cases.

1. $R=k[[X_1, X_2, \ldots, X_n]]$.
2. $R=k[X_1, X_2, \ldots, X_n]_{(X_1, X_2, \ldots, X_n)}$.

Note that if $R/P$ is regular then $P$ is generated by a sequence. If $dimR=2$ then $P$ is principal . So in either cases $P^2$ is $P$ primary. In this problem we have to prove or disprove that $\mathfrak{m} \notin Ass R /P^2$ or $R/P^2$ is one dimensional cohen macaulay ring.

This is false in general. If $$R$$ is a $$3$$ dimensional regular local ring and $$P$$ is a height $$2$$ prime ideal which is not a complete intersection then $$P^n \neq P^{(n)}$$ for $$n\geq 2$$ which is equivalent to saying none of the powers of $$P$$ other than $$P$$ itself are $$P$$-primary.