A property of weakly Krull domains

We show that a weakly Krull domain $D$ satisfies $(\ast )$: for every pair $a,b\in D\backslash \{0\}$ there is an $n=n(a,b)\in \mathbb{N}$ such that $(a,b^{n})$ is $t$-invertible. For $D$ Noetherian, $D$ satisfies $(\ast )$ if and only if every grade-one prime ideal of $D$ is of height one. We also show that a modification of $(\ast )$ can be used to characterize Noetherian domains that are one-dimensional.