Real closed rings arise in semi-algebraic geometry and topology as well as in the investigation of partially ordered rings. It is shown that localizations of real closed rings with respect to Gabriel filters, or more generally: multiplicative filters, are again real closed. Thus, real closedness is preserved under a large number of important ring theoretic constructions. For a few particularly simple cases the multiplicative filters are classified and the localizations are determined.
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Niels Schwartz, Gabriel filters in real closed rings. Comment. Math. Helv. 72 (1997), no. 3, pp. 434–465DOI 10.1007/S000140050027