Sv-rings and Sv-porings

SV-rings are commutative rings whose factor rings modulo prime ideals are valuation rings. SV-rings occur most naturally in connection with partially ordered rings (= porings) and have been studied only in this context so far. The present note first develops the theory of SV-rings systematically, without assuming the presence of a partial order. Particular attention is paid to the question of axiomatizability (in the sense of model theory). Partially ordered SV-rings (SV-porings) are introduced, and some elementary properties are exhibited. Finally, SV-rings are used to study convex subrings and convex extensions of porings, in particular of real closed rings. An SV-ring is a commutative ring whose factor rings modulo prime ideals are always valuation rings. Originally the notion was introduced and studied in connection with rings of continuous functions and f-rings (cf. [15], [16], [13], [14]). In the introduction of [14] the authors noted that there is no reason why the study of these rings should be restricted a priori to partially ordered rings (p. 195). The present note starts with the study of SV-rings without partial orders. The second part of the paper deals with SV-rings that are also porings. The prime spectrum of an SV-ring is completely normal, which means that the specializations of a prime ideal always form a chain with respect to inclusion (or, specialization). Thus, SV-rings only occur in situations where there are "many" minimal prime ideals. This means: If there is any set M ! Spec A ( ) of mutually incomparable prime ideals then there at least as many distinct minimal prime ideals. Most rings in classical number theory or algebraic geometry, e.g., Noetherian rings, do not have this property. On the other hand, such a property is not uncommon in real algebra, e.g., in rings of continuous functions, real closed rings and f-rings. From this perspective it seems rather natural that SV-rings have been studied exclusively in real algebra so far. Valuation rings have a long and distinguished history; there are a large number of standard texts about the subject, [7] being the most recent one (where further references can be looked up). Cherlin and Dickmann were first to ask for factor domains of rings of continuous functions that are valuation rings ([4], [5]). Most factor domains of rings of continuous functions are not valuation rings; if they are valuation rings then they are convex subrings of real closed fields. Cherlin and Dickmann called such rings real closed; here they will be called real closed valuation rings. [35] is a study of real closed valuation rings vs. the larger class of real closed domains. Henriksen and Wilson continued this line of investigation by asking for topological spaces for which every factor domain of the ring of continuous functions is a valuation ring ([15], [16]). Such spaces are called SV-spaces; their rings of continuous functions are called SV-rings. One class of SV-spaces has been known for a long time: F-spaces ([8], 14.25). These spaces arise naturally in connection with the Stone-Cech compactification. Every zero set of !X that does not meet X is an F-space ([8], 14O). There exist other SVspaces ([16]), but so far they seem to be artificial constructs and it is not clear whether they arise anywhere in a natural way. Later the notion of SV-rings was extended to f-rings ([13]). More studies of SV-rings may be found in [14], [24], [25] and [32]. A completely different class of SV-rings arises in semi-algebraic geometry: The ring of continuous semialgebraic functions on a real algebraic curve is an SV-ring ([35], Corollary 2.6). This is a significant fact since the Curve Selection Lemma ([6], § 12) can be used to reduce many questions about semi-algebraic functions to questions about functions defined on curves. There is a potential for applications of SV-rings in semi-algebraic geometry. The present note starts with a definition and various examples of SV-rings (without partial orders). Basic properties of SV-rings are explored in section 1 and section 3. Section 3 is mostly devoted to the question whether the class of all SV-rings, or some significant subclasses, is elementary (or axiomatizable) with respect to the language of ring theory (cf. [3], p. 173, or [18], p. 34). Reduced SV-rings are characterized by the fact that every pair of elements satisfies some polynomial identity (Theorem 3.4). The identity, in particular its degree, depends on the pair of elements. For axiomatizability of the class of SV-rings one would need a uniform bound on the degree of the identities. The degree of the polynomial identities is closely connected with the rank of the ring. The rank is defined to be the supremum of the numbers of minimal prime ideals that are contained in a single maximal ideal. It has been shown in [14] that a ring of