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.,
. 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 language ...
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 ra ...
s, such as the ring of
dyadic rational
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in compu ...
s.
A typical example is given by
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is af ...
: 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 fin ...
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 finitel ...
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
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