  We extend a result of Minh and Trung to get criteria for $\depth
I=\depth\sqrt{I}$ where $I$ is an unmixed monomial ideal of the polynomial ring
$S=K[x_1,..., x_n]$. As an application we characterize all the pure simplicial
complexes $\Delta$ which have rigid depth, that is, which satisfy the condition
that for every unmixed monomial ideal $I\subset S$ with $\sqrt{I}=I_\Delta$ one
has $\depth(I)=\depth(I_\Delta).$
