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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 anuntilting map $(-)^{\sharp}$ from the set of radical ideals of $R^{\flat}$ tothe set of ideals of $R$. The untilting map $(-)^{\sharp}$ is defined purelyalgebraically and generalizes the analytically defined untilting map on closedradical ideals of a perfectoid Tate ring of characteristic $p$ introduced bythe first author. We prove that these two maps, $(-)^{\flat}$ and$(-)^{\sharp}$, define an inclusion-preserving bijection between the set ofideals $J$ of $R$ such that the quotient $R/J$ is perfectoid and the set of$p^{\flat}$-adically closed radical ideals of $R^{\flat}$, where $p^{\flat}\inR^{\flat}$ corresponds to a compatible system of $p$-power roots of a unitmultiple of $p$ in $R$. Furthermore, we prove that the maps send (closed) primeideals to prime ideals and thus define a homeomorphism between the subspace ofthe 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 $R^{\flat}$. Inparticular, we obtain a generalization and a new proof of the main result ofthe first author's previous research which concerned prime ideals in perfectoidTate rings.

Tilting and untilting for ideals in perfectoid rings