Boolean families of valuation rings. II (Q1346896)

The aim of the paper is to strengthen the author's earlier results [Algebra Logic 31, No. 3, 170-181 (1992); translation from Algebra Logika 31, No. 3, 276-296 (1992; Zbl 0791.12004)]. It is shown that if \(W\) is a weakly Boolean family of valuation rings of a field \(F\) and \(R= R_ W \rightleftharpoons \cap \{R_ V\mid R_ V\in W\}\) is a Prüfer ring with field of quotients \(F\), then \(W\) is Boolean. It is a considerably stronger version of proposition 4 of the above-cited paper. Sufficient conditions for a family of valuation rings of a field to be Boolean are formulated. It is proved that every regular Prüfer ring is a Bezout ring.