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Extending work of Puninski, Puninskaya and Toffalori in [5], we show that if V is an effectively given valuation domain then the theory of all V-modules is decidable if and only if there exists an algorithm which, given a, b ε V, answers whether a ε rad(bV). This was conjectured in [5] for valuation domains with dense value group, where it was proved for valuation domains with dense archimedean value group. The only ingredient missing from [5] to extend the result to valuation domains with dense value group or infinite residue field is an algorithm which decides inclusion for finite unions of Ziegler open sets. We go on to give an example of a valuation domain with infinite Krull dimension, which has decidable theory of modules with respect to one effective presentation and undecidable theory of modules with respect to another. We show that for this to occur infinite Krull dimension is necessary.

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For Bézout domains these conditions are also necessary. IntroductionWe deal here with decidability of first order theories of modules over Prüfer (in particular Bézout) domains R with infinite residue fields. We assume R effectively given (so countable), in order to ensure that the decision problem for R-modules makes sense.The model theory of modules over Bézout domains, with some hints at Prüfer domains, is studied in [16]. The decidability of the theory of modules over the ring of algebraic integers is proved in [10] (see also [8]), and a similar result is obtained in [15] over Bézout domains obtained from principal ideal domains by the so called D+M-construction [3].On the other hand Gregory [5], extending [14], proved that the theory of modules over a(n effectively given) valuation domain V is decidable if and only if there is an algorithm which decides the prime radical relation, namely, for every a, b ∈ V , answers whether a ∈ rad(bV ) (equivalently, whether the prime ideals of V containing b also include a).This paper develops a similar analysis in a closely related setting, that is, over Prüfer domains.In fact a domain is Prüfer if and only if all its localizations at maximal ideals are valuation domains. Bézout domains are a notable subclass of Prüfer domains. In both cases we focus on the domains all of whose residue fields with respect to maximal ideals are infinite. The reason and the benefit of this choice are illustrated in § 3 below. Notice that Prüfer (indeed Bézout) domains with infinite residue fields include the ring of algebraic integers and the ring of complex valued entire functions -even if the latter is uncountable and so cannot be effectively given (but see the analysis of its Ziegler spectrum in [9]). Other noteworthy examples will be proposed in § 6.2000 Mathematics Subject Classification. 03C60 (primary), 03C98, 03B25, 13F05. while this paper was being completed. The other three authors would like to dedicate this article as a tribute to his memory. 1Our main result, resembling [5], states that, if R is such a Bézout domain, then the theory of R-modules is decidable if and only if there is an algorithm which answers a sort of double prime radical relation, in detail, given a, b, c, d ∈ R, decides whether, for all prime ideals p, q with p+q = R, b ∈ p implies a ∈ p or d ∈ q implies c ∈ q. This will be proved in § 6. Generalizations to Prüfer domains will be presented in the final part of the paper, in § 7. The preceding sections § § 2-5 describe the framework of (effectively given) Prüfer domains and prepare the main theorems.We refer to all the already mentioned papers and books, as well as to the key references on model theory of modules, [11], [12] and [18]. We also assume some familiarity with Prüfer domains, as treated, for instance, in [3] an...

We study the R‐torsion‐free part of the Ziegler spectrum of an order Λ over a Dedekind domain R. We underline and comment on the role of lattices over Λ. We describe the torsion‐free part of the spectrum when Λ is of finite lattice representation type.

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains whose localizations at maximal ideals have dense value groups. For Bézout domains, these conditions are also necessary.2010 Mathematics Subject Classification. 03C60 (primary), 03C98, 03B25, 13F05.