  We consider the ideal structure of a reduced crossed product of a unital
$C^*$-algebra equipped with an action of a discrete group. More specifically we
find sufficient and necessary conditions for the group action to have the
intersection property, meaning that non-zero ideals in the reduced crossed
product restrict to non-zero ideals in the underlying $C^*$-algebra. We show
that the intersection property of a group action on a $C^*$-algebra is
equivalent to the intersection property of the action on the equivariant
injective envelope. We also show that the centre of the equivariant injective
envelope always contains a $C^*$-algebraic copy of the equivariant injective
envelope of the centre of the injective envelope. Finally, we give applications
of these results in the case when the group is $C^*$-simple.
