# 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., .[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.[2] 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.

## References

- Fontana, Marco; Papick, Ira J. (2002), "Dedekind and Prüfer domains", in Mikhalev, Alexander V.; Pilz, Günter F. (eds.),
*The concise handbook of algebra*, Kluwer Academic Publishers, Dordrecht, pp. 165–168, ISBN 9780792370727. - Fuchs, Laszlo; Heinzer, William; Olberding, Bruce (2004), "Maximal prime divisors in arithmetical rings",
*Rings, modules, algebras, and abelian groups*, Lecture Notes in Pure and Appl. Math.,**236**, Dekker, New York, pp. 189–203, MR 2050712. See in particular p. 196.

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