A property of weakly Krull domains

Abstract

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