These notes give a careful study of relative Prüfer rings and Manis valuations with an eye towards applications to real and \(p\)-adic algebraic geometry. The main topic is Prüfer ring extensions, where an extension \(A\subset R\) of commutative rings is called Prüfer if \(A\) is \(R\)-Prüfer in the sense of M. Griffin that \((A_{[P]}, P_{[P]})\) is a Manis pair in \(R\) for every maximal ideal \(P\) of \(A\).
This volume has three chapters. The first chapter develops the basic properties of Manis valuations and Prüfer extensions; the second chapter studies Prüfer extensions from a ideal-theoretic rather than a valuation point of view; and the final chapter studies several special types of Manis valuations. As indicated by the title, a second volume is also planned.