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Bezout and Prüfer f-rings. (English) Zbl 0766.06018

Authors’ summary: This article describes Bezout and Prüfer \(f\)-rings in terms of their localizations. All \(f\)-rings here are commutative, semiprime and possess an identity; they also have the bounded inversion property: \(a>1\) implies that \(a\) is a multiplicative unit. The two main theorems are as follows: (1) \(A\) is a Bezout \(f\)-ring if and only if each localization at a maximal ideal is a (totally ordered) valuation ring; (2) Each Prüfer \(f\)-ring is quasi-Bezout, and if each localization of \(A\) is a Prüfer \(f\)-ring, then so is \(A\). We give a counterexample to show that the converse of the last assertion is false.

MSC:

06F25 Ordered rings, algebras, modules
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References:

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