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., $A\subseteq B\subseteq K$ .^[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⁻¹A 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., 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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[1] Fontana, Marco; Papick, Ira J. (2002), "Dedekind and Prüfer domains", in Mikhalev, Alexander V.; Pilz, Günter F., The concise handbook of algebra, Kluwer Academic Publishers, Dordrecht, pp. 165–168, ISBN 9780792370727 .

[2] 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.