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Functorial rings of quotients. II: The ring of hyper-fractions. (English) Zbl 0804.18002

Do not think of reading this paper before reading its predecessor [Proc. Conf. Ordered algebraic structures, Gainesville 1991, pp. 133-157 (1993; Zbl 0792.06017)]. The authors push on toward maximal functorial rings of quotients. Indeed, they mention that a future paper will establish such extensions for Archimedean \(f\)-rings. The burden of this paper is the subring \(A^ H\) of the maximal ring of quotients consisting of all fractions whose domain is “absolutely dense”: having dense image under every morphism. The setting is still a remarkably general category over (commutative) rings (with 1). \(A \to A^ H\) is always an epimorphism, and \(A \mapsto A^ H\) is a ring-of-quotients functor. However, it tends not to be a classical ring of quotients unless it coincides with the simpler \(A_ H\) studied in the preceding paper. Among the results to this effect one is that it suffices that the kernels of morphisms in the category (i) include all annihilator ideals and (ii) are closed under filtered union, and (iii) every finitely generated hyper-dense ideal has a hyper-regular element. The further results are more technical.

MSC:

18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
06F25 Ordered rings, algebras, modules
13B30 Rings of fractions and localization for commutative rings
13A99 General commutative ring theory

Citations:

Zbl 0792.06017
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