Bezout and prufer f-rings

This article describes Bezout and Prüfer f-rings in terms of their localizations. All f-rings here are corrmutative, semi prime 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 Prufer f-ring is quasi-Bezout, and if each localization of A is a Prufer f-ring then so is A. We give a counter-example to show that the converse of the last assertion is false.