  When $E$ is an $R$-module over a commutative unital ring $R$, the Zariski
closure of its support is of the form $\mathrm V(\mathcal O(E))$ where
$\mathcal O(E)$ is a unique radical ideal. We give an explicit form of
$\mathcal O(E)$ and study its behavior under various operations of algebra.
Applications are given, in particular for ring extensions of commutative unital
rings whose supports are closed. We provide some applications to crucial and
critical ideals of ring extensions.
