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The paper is a continuation of work initiated by the first two authors in [K-M]. Section 1 is introductory. In Section 2 we prove a basic lemma, Lemma 2.1, and use it to give new proofs of key technical results of Scheiderer in [S1] [S2] in the compact case; see Corollaries 2.3, 2.4 and 2.5. Lemma 2.1 is also used in Section 3 where we continue the examination of the case n = 1 initiated in [K-M], concentrating on the compact case. In Section 4 we prove certain uniform degree bounds for representations in the case n = 1, which we then use in Section 5 to prove that (‡) holds for basic closed semi-algebraic subsets of cylinders with compact cross-section, provided the generators satisfy certain conditions; see Theorem 5.3 and Corollary 5.5. Theorem 5.3 provides a partial answer to a question raised by Schmüdgen in [Sc2]. We also show that, for basic closed semi-algebraic subsets of cylinders with compact cross-section, the sufficient conditions for (SMP) given in [Sc2] are also necessary; see Corollary 5.2(b). In Section 6 we prove a module variant of the result in [Sc2], in the same spirit as Putinar's variant [Pu] of the result in [Sc1] in the compact case; see Theorem 6.1. We apply this to basic closed semi-algebraic subsets of cylinders with compact cross-section; see Corollary 6.4. In Section 7 we apply the results from Section 5 to solve two of the open problems listed in [K-M]; see Corollary 7.1 and Corollary 7.4. In Section 8 we consider a number of examples in the plane. In Section 9 we list some open problems.

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We investigate connections between arithmetic properties of rings and topological properties of their prime spectrum. Any property that the prime spectrum of a ring may or may not have, defines the class of rings whose prime spectrum has the given property. We ask whether a class of rings defined in this way is axiomatizable in the model theoretic sense. Answers are provided for a variety of different properties of prime spectra, e.g., normality or complete normality, Hausdorffness of the space of maximal points, compactness of the space of minimal points.

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The relationship between a poring and a convex subring has been studied most successfully for porings with bounded inversion. The best conceivable results are known for real closed rings. The present paper focuses on the connections between the prime spectra and between the real spectra of a poring and a convex subring. Examples show that for arbitrary porings one may not expect a very close relationship. But with the assumption of bounded inversion or spectral compatibility the results are similar to those for real closed rings.