A noetherian local ring having a height one principal prime is a domain

Question

$A$ is a commutative ring with with $1$. If $A$ is a Noetherian and local ring and $A$ has a principal prime ideal of height $1$ then show that $A$ is a domain.

Can anybody give some hint.I tried to show $(0)$ is a prime ideal in $A$ but I couldn't.

Answer 1

Let $\mathfrak p=(p)$ be a principal prime of height one. Then there is a prime $\mathfrak q\subsetneq\mathfrak p$. If $\mathfrak q=(0)$ you are done. Otherwise, let $a\in\mathfrak q$, $a\ne 0$. Then $a=pa_1$ and since $p\notin\mathfrak q$ it follows that $a_1\in\mathfrak q$, so $a_1=pa_2$, and so on. Thus $a\in\bigcap_{n\ge0}\mathfrak p^n\subseteq\bigcap_{n\ge0}\mathfrak m^n=(0)$, a contradiction. (Here $\mathfrak m$ denotes the maximal ideal.)