Convexity, valuations and Prüfer extensions in real algebra
In the monograph [\textit{M. Knebusch} and \textit{D. Zhang}, Manis valuations and Prüfer extensions. I: A new chapter in commutative algebra. Lect. Notes Math. 1791 (2002; Zbl 1033.13001)] the authors developed a theory of Manis valuations on rings and Prüfer extensions. In the present paper they study these notions in the context of real algebra, i.e., in the presence of some form of order structure (preorderings, partial orderings, total orders,\dots). Convex subrings of totally ordered fields are well known to be valuation rings. Therefore valuations arise naturally in the theory of ordered rings, in particular in connection with convexity. (A subset \(M\) of a preordered ring \((A,A^+)\) is convex if \(a, b\in M\) and \(c- a, b- c\in A^+\) implies \(c\in M\).) Throughout, the paper is concerned with convexity and its relationship with valuations. Ordered rings and convex subrings lead to examples of Prüfer extensions and Prüfer closed rings. In the other direction, Prüfer theory is a powerful tool in the study of convex subrings The notion of a convex valuation on a ring generalizes the classical concept of compatibility of a valuation on a field with a total order. In most cases this means that the corresponding valuation ring is convex. These are the valuations that are useful in the study of ordered rings. The relationship between intersections of convex valuation rings, notably holomorphy rings, and convex subrings is one main topic. These concepts are studied quite generally in the beginning. The most substantial results are obtained with additional assumptions about the order (such as bounded inversion: \(1+ A^+\subseteq A^\times\)), or about the ring (e.g., \((A, A^+)\) is an \(f\)-ring), or both. If bounded inversion holds then: \(A\) is a Prüfer extension of every convex subring, say \(B\); every intermediate ring of \(B\) and \(A\) is convex in \(A\) as well and is a ring of quotients of \(B\). Even without bounded inversion it is shown that for any Prüfer subring \(B\subseteq A\) there is a largest intermediate ring that contains \(B\) as a convex subring. Stronger versions of these results hold for \(f\)-rings. The results are applied to rings of continuous functions; it is shown that rings of continuous functions on metric spaces are Prüfer closed. Finally, valuations on \(f\)-rings are studied.
scientific article; zbMATH DE number 2146818