Gabriel filters in real closed rings (Q1379592)

The paper consists of seven sections: 1. \(l\)-ideals in real closed rings; 2. Multiplicative filters in arbitrary rings; 3. Deligne's formula; 4. Filters of finite type in real closed rings; 5. Filters in real closed domains; 6. Localization of real closed rings are real closed rings; 7. Multiplicative filters and the Keimel spectrum. The author investigates Gabriel filters, multiplicative filters, filters of finite type and spaces of prime ideals of real closed rings [see \textit{N. Schwartz}, in: Algebra and Order, Proc. 1st Int. Symp., Luminy 1984, Res. Expo. Math. 14, 175-194 (1986; Zbl 0616.14019)]. The class of real closed rings contains rings of continuous real functions on topological spaces. It is established that for a real closed ring \(A\) the following three sets have the same cardinality: (i) the set of isomorphic classes of flat epimorphisms over \(A\); (ii) the set of Gabriel filters of finite type; (iii) the set of generically closed proconstructible subsets of \(\text{Spec}(A)\). It is proved that in a real closed domain every multiplicative filter is a Gabriel filter and is given a description of filters of finite type in real closed domains. The main results of the paper are contained in section 6 where is proved that if \(A\) is a real closed ring and \(\mathcal F\) a Gabriel filter in \(A\) then its localization \(L_{\mathcal F}(A)\) is real closed. It is obtained that the complete ring of quotients \(Q(A)\) of a real closed ring \(A\) is real closed.