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In mathematics, a real closed ring is a commutative ring ''A'' that is a subring of a product of real closed fields, which is closed under continuous semi-algebraic functions defined over the integers. == Examples of real closed rings == Since the rigorous definition of a real closed ring is of technical nature it is convenient to see a list of prominent examples first. The following rings are all real closed rings: * real closed fields. These are exactly the real closed rings that are fields. * the ring of all real valued continuous functions on a completely regular space ''X''. Also, the ring of all bounded real valued continuous functions on ''X'' is real closed. * convex subrings of real closed fields. These are precisely those real closed rings which are also valuation rings and were initially studied by Cherlin and Dickmann (they used the term 'real closed ring' for what is now called 'real closed valuation ring'). * the ring ''A'' of all continuous semi-algebraic functions on a semi-algebraic set of a real closed field (with values in that field). Also, the subring of all bounded (in any sense) functions in ''A'' is real closed. * (generalizing the previous example) the ring of all (bounded) continuous definable functions on a definable set ''S'' of an arbitrary first-order expansion ''M'' of a real closed field (with values in ''M''). Also, the ring of all (bounded) definable functions is real closed. * Real closed rings are precisely the rings of global sections of affine real closed spaces (a generalization of semialgebraic spaces) and in this context they were invented by Niels Schwartz in the early 1980s. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「real closed ring」の詳細全文を読む スポンサード リンク
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