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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., .〔.〕 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''−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 particular (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==
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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