  For an (integral) perfectoid ring $R$ of characteristic $0$ with tilt
$R^{\flat}$, we introduce and study a tilting map $(-)^{\flat}$ from the set of
$p$-adically closed ideals of $R$ to the set of ideals of $R^{\flat}$ and an
untilting map $(-)^{\sharp}$ from the set of radical ideals of $R^{\flat}$ to
the set of ideals of $R$. The untilting map $(-)^{\sharp}$ is defined purely
algebraically and generalizes the analytically defined untilting map on closed
radical ideals of a perfectoid Tate ring of characteristic $p$ introduced by
the first author. We prove that these two maps, $(-)^{\flat}$ and
$(-)^{\sharp}$, define an inclusion-preserving bijection between the set of
ideals $J$ of $R$ such that the quotient $R/J$ is perfectoid and the set of
$p^{\flat}$-adically closed radical ideals of $R^{\flat}$, where $p^{\flat}\in
R^{\flat}$ corresponds to a compatible system of $p$-power roots of a unit
multiple of $p$ in $R$. Furthermore, we prove that the maps send (closed) prime
ideals to prime ideals and thus define a homeomorphism between the subspace of
the spectrum of $R$ consisting of prime ideals $\mathfrak{p}$ of $R$ such that
$R/\mathfrak{p}$ is perfectoid and the subspace of the spectrum of $R^{\flat}$
consisting of $p^{\flat}$-adically closed prime ideals of $R^{\flat}$. In
particular, we obtain a generalization and a new proof of the main result of
the first author's previous research which concerned prime ideals in perfectoid
Tate rings.
