  Given a free resolution of an ideal $\mathfrak a$ of holomorpic functions
there is an associated residue current $R$ that coincides with the classical
Coleff-Herrera product if $\mathfrak a$ is a complete intersection ideal and
whose annihilator ideal equals $\mathfrak a$. In the case when $\mathfrak a$ is
an Artinian monomial ideal, we show that the singularities of $R$ are small in
a certain sense if and only if $\mathfrak a$ is integrally closed.
