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In mathematics, an overring ''B'' of an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
''A'' is a subring of the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
''K'' of ''A'' that contains ''A'': i.e., A \subseteq B \subseteq K. For instance, an overring of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s is a ring in which all elements are
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s, such as the ring of dyadic rationals. A typical example is given by localization: if ''S'' is a
multiplicatively closed subset In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold: * 1 \in S, * xy \in S for all x, y \in S. In other words, ''S'' is closed under taking finite ...
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 domain In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every fini ...
s and are a subset of the
Prüfer domain In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely gen ...
s.. See in particula
p. 196
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 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
.


References

Ring theory {{Abstract-algebra-stub