On $\frak{m}$-Full Powers of Parameter Ideals

Abstract

Let $Q$ be a parameter ideal in a Noetherian local ring $A$ with the maximal ideal $\frak{m}$. Then $A$ is a regular local ring and $\frak{m}/Q$ is cyclic, if $\rm{depth}\ A > 0$ and $Q^n$ is $\frak{m}$-full for some integer $n \geq 1$. Consequently, $A$ is a regular local ring and all the powers of $Q$ are integrally closed in $A$ once $Q^n$ is integrally closed for some $n \geq 1$.