  Let $B \subseteq A$ be an inclusion of C$^*$-algebras. We study the
relationship between the regular ideals of $B$ and regular ideals of $A$. We
show that if $B \subseteq A$ is a regular C$^*$-inclusion and there is a
faithful invariant conditional expectation from $A$ onto $B$, then there is an
isomorphism between the lattice of regular ideals of $A$ and invariant regular
ideals of $B$. We study properties of inclusions preserved under quotients by
regular ideals. This includes showing that if $D \subseteq A$ is a Cartan
inclusion and $J$ is a regular ideal in $A$, then $D/(J\cap D)$ is a Cartan
subalgebra of $A/J$. We provide a description of regular ideals in reduced
crossed products $A \rtimes_r \Gamma$.
