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.,
. For instance, an overring of the
integers is a ring in which all elements are
rational numbers, 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 compute ...
s.
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.
[. 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.
References
Ring theory
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