Manis valuations and Prüfer extensions. I (Q1917630)

This paper should be read by everyone interested in valuation theory and its applications to real and \(p\)-adic algebraic geometry. Let \(A \subset R\) be a (commutative with 1) ring extension. \(A\) is called an \(R\)-Prüfer ring if the saturated pair \((A_{[p]}, p_{[p]})\) (i.e., the pair of \((A \backslash p)\)-components of \((A,p))\) is a Manis pair in \(R\) for every maximal ideal \(p\) of \(A\), namely if there exists a Manis valuation \(v\) of \(R\) such that \(A_{[p]} = \{x \in R |v(x) \geq 0\}\) and \(p_{[p]} = \{x \in R |v(x) > 0\}\). This definition, which improves the one given by \textit{M. Griffin} [Canad. J. Math. 26, 412-429 (1974; Zbl 0259.13008)], establishes the central notion of the paper. The relation between Prüfer domains and valuations of their quotient fields has a lot of interest in real and \(p\)-adic semi-algebraic geometry. On the other hand, if \(A \subset R\) is a (commutative with 1) ring extension, we can define a real holomorphy ring consisting of those elements of \(R\) which on the real spectrum of \(R\) can be bounded by elements of \(A\). This ring is also very useful in real semi-algebraic geometry and with suitable conditions (given in the paper) is an \(R\)-Prüfer ring. This motivates the following observation: Since an \(R\)-Prüfer ring is related to Manis valuations of \(R\) in much the same way as a Prüfer domain is related to valuations of its quotient field, the study of \(R\)-Prüfer ring and Manis valuations could allow to obtain interesting consequences in the field of semi-algebraic geometry. This study is the final objective of this interesting paper. Actually, the article is only the first chapter of a book in preparation. In the paper, a very careful investigation about a foundation basic theory of \(R\)-Prüfer rings and Manis valuations (with an eye to applications in geometry) can be found. Even though the idea of the above described program has been just followed (in a different way) by \textit{M. Marshall} [J. Algebra 140, No. 2, 484-501 (1991; Zbl 0752.13002)] very little seems to be done on this subject. First the paper, inspired by \textit{P. Roquette}'s one [``Principal ideal theorems for holomorphy rings in fields'', J. Reine Angew. Math. 262/263, 361-374 (1973; Zbl 0271.13009)] recollects all the properties about Manis valuations which will be needed in the rest of it. Afterwards, a theory about certain ring homomorphisms called weakly surjectives is developed. The main part of the paper is section 5 where the authors relate the above results characterizing Prüfer extensions in terms of weak surjectivity and moreover they obtain many more characterizations of Prüfer extensions. Finally, various examples of Manis valuations and Prüfer extensions are given.