Overring

In mathematics, an overring "B" of an integral domain "A" is a subring of the field of fractions "K" of "A" that contains "A": i.e., formula_1. For instance, an overring of the integers is a ring in which all elements are rational numbers, such as the ring of dyadic rationals.
A typical example is given by localization: if "S" is a multiplicatively closed subset of "A", then the localization "S"−1"A" is an overring of "A". The rings in which every overring is a localization are said to have the QR property; they include the Bézout domains and are a subset of the Prüfer domains. In particular, every overring of the ring of integers arises in this way; for instance, the dyadic rationals are the localization of the integers by the powers of two.