Real Prüfer extensions of commutative rings (Q5957332)

Let \(A\subset R\) be an extension of commutative rings with 1. A is called totally real if \(1+\sum_i a_i^2\) is a unit in \(A\) for all finite collections \(\{a_1,\ldots, a_n\}\subset A.\) The set \[ H(R/A)=\{x\in R: \exists a\in A, -a\leq x \leq a \text{ on Sper } R\} \] is called the real holomorphy ring of \(R\) over \(A\). Main result: \(A\) is totally real and \(A\subset R\) a Prüfer extension if and only if \(R\) is totally real and \(A=H(R/A).\) The proof relies heavily on papers by \textit{M. Knebusch} and \textit{D. Zhang} [Doc. Math., J. DMV 1, 149-198 (1996; Zbl 0855.13001)] and by \textit{N. Schwartz} [Manuscr. Math. 102, 347-381 (2000; Zbl 0966.13018)].