Overring

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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[edit]

  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 .
  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.