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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.

We study the R-torsionfree part of the Ziegler spectrum of an order \Lambda over a Dedekind domain R. We underline and comment on the role of lattices over \Lambda. We describe the torsionfree part of the spectrum when \Lambda is of finite lattice representation type.

We prove that the theory of all modules over the ring of algebraic integers is decidable., Comment: 13 pages

We will describe the Ziegler spectrum over the ring of entire complex valued functions.

We wander through logic and justice, theorems and trials, reporting several cases of often imaginary legal proceedings and court cases in which logic is either a useful tool, a paradoxical presence, or an embarrassing absence. We call as witnesses Dostoyevsky, Carroll, Cervantes, Diderot, Chesterton, Kafka and many others. We also call a hasty trial against logic itself and its supposed benefits for research in mathematics, which ends in an equally hasty acquittal.

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prufer (in particular Bezout) domains whose localizations at maximal ideals have dense value groups. For Bezout domains, these conditions are also necessary.

The image of mathematics in Fyodor Dostoyevsky’s novels is clearly negative. Notes from Underground contain several famous and violent pages against Mathematics, and this criticism is also echoed in other masterpieces, such as Crime and Punishment and The Possessed. Dostoevsky explicitly accuses mathematical determinism of being arrogant and oppressive. On the other hand, some of his main characters, such as Kirillov in The Possessed and Ivan in The brothers Karamazov, seem to disclose unexpected mathematical sympathies. Here, in Part II of a two-part article, we discuss Dostoyevsky’s opinion of mathematics and, more generally, of science. We also compare truth and freedom in mathematics and in the vision of the Russian writer.

We consider Hilbert’s contributions to the foundations of mathematics, including his conception of the axiomatic method, his notion of mathematical proof, and his approach to infinity through finitary tools. We describe his “Programme”, and we comment its collapse due to Godel’s Incompleteness Theorems. Despite that defeat, Hilbert’s project remains topical today.

Let R be a pullback of Dedekind domains onto a common quotient field. I discuss the pure injectivity of certain R-modules arising in a natural way as generalizations of indecomposable pure injective finitely generated R-modules.

We will describe the torsion-free part of the Ziegler spectrum, both the points and the topology, over the integral group ring of the Klein group. For instance we will show that the Cantor–Bendixson rank of this space is equal to 3.

Keywords. Local o-minimality, Strong local o-minimality, !-categoricity. Abstract. We introduce and study some local versions of o-minimality, requir- ing that every definable set decomposes as the union of finitely many isolated points and intervals in a suitable neighborhood of every point. Motivating ex- amples are the expansions of the order of reals by sine, cosine and other periodic functions.

On the basis of the Klingler–Levy classification of finitely generated modules over commutative noetherian rings we approach the old problem of classifying finite commutative rings R with a decidable theory of modules. We prove that if R is (finite length) wild, then the theory of all R -modules is undecidable, and verify decidability of this theory for some classes of tame finite commutative rings.

We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are "tame" according to the Klingler–Levy analysis in [4–6], in particular over Dedekind-like rings and their homomorphic images. We show that both are undefined in most cases.

We investigate several model theoretic minimalities in the framework of modules over a given serial ring R.

We classify group rings of finite groups over a field F according to the model-theoretic complexity of the category of their modules. For instance, we prove that, if F contains a primitive cubic root of 1, then the Krull�Gabriel dimension of such rings is 0, 2, or undefined.

We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all V -modules is decidable.

We will develop the model theory of modules over commutative Bezout domains. In particular we characterize commutative Bezout domains B whose lattice of pp-formulae has no width and give some applications to the existence of superdecomposable pure injective B-modules.

We are interested in (right) modules M satisfying the following weak divisibility condition: If R is the underlying ring, then for every r ∈ R either Mr = 0 or Mr = M. Over a commutative ring, this...

We prove that if G is a non-trivial finite group, then the integral group ring $\mathbb{Z} G$ possesses a superdecomposable pure-injective module.

A module is weakly minimal if and only if every pp-definable subgroup is either finite or of finite index. We study weakly minimal modules over several classes of rings, including valuation domains, Prufer domains and integral group rings. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We discuss Vaught's conjecture for complete theories of modules over some group rings over the integers and other Dedekind domains, and over pullback rings of Dedekind domains.

The study of pairs of modules (over a Dedekind domain) arises from two different perspectives, as a starting step in the analysis of tuples of submodules of a given module, or also as a particular case in the analysis of Abelian structures made by two modules and a morphism between them. We discuss how these two perspectives converge to pairs of modules, and we follow the latter one to obtain an alternative approach to the classification of pairs of torsionfree objects. Then we restrict our attention to pairs of free modules. Our main results are that the theory of pairs of free Abelian groups is co-recursively enumerable, and that a few remarkable extensions of this theory are decidable.

We consider modules over a group ring RG where R is a countable Dedekind domain and G is a finite group. We describe the internal structure of those RG-modules which are strongly minimal or satisfy other related model theoretic and algebraic minimality conditions.

We introduce and discuss the recent Agrawal–Kayal–Saxena deterministic algorithm solving primality in polynomial time.

Discrete weakly o-minimal structures, although not so stimulating as their dense counterparts, do exhibit a certain wealth of examples and pathologies. For instance they lack prime models and monotonicity for definable functions, and are not preserved by elementary equivalence. First we exhibit these features. Then we consider a countable theory of weakly o-minimal structures with infinite definable discrete (convex) subsets and we study the Boolean algebra of definable sets of its countable models. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We study ω-categorical weakly o-minimal expansions of Boolean lattices. We show that a structure = (A,≤, ℐ) expanding a Boolean lattice (A,≤) by a finite sequence I of ideals of A closed under the usual Heyting algebra operations is weakly o-minimal if and only if it is ω-categorical, and hence if and only if A/I has only finitely many atoms for every I ∈ ℐ. We propose other related examples of weakly o-minimal ω-categorical models in this framework, and we examine the internal structure of these models.

We consider R-torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. In the first part of the paper [4] we compared the theory T(RG) of all R-torsionfree RG-modules and the theory T0(RG) of RG-lattices (i. e. finitely generated R-torsionfree RG-modules), and we realized that they are almost always different. Now we compare their behaviour with respect to decidability, when RG-lattices are of finite, or wild representation type.

For every ring S with identity, the (right) Ziegler spectrum of S, Zgs, is the set of (isomorphism classes of) indecomposable pure injective (right) S-modules. The Ziegler topology equips Zgs with the structure of a topological space. A typical basic open set in this topology is of the formwhere φ and ψ are pp-formulas (with at most one free variable) in the first order language Ls for S-modules; let [φ/ψ] denote the closed set Zgs - (φ/ψ). There is an alternative way to introduce the Ziegler topology on Zgs. For every choice of two f.p. (finitely presented) S-modules A, B and an S-module homomorphism f: A → B, consider the set (f) of the points N in Zgs such that some S-homomorphism h: A → N does not factor through f. Take (f) as a basic open set. The resulting topology on Zgs is, again, the Ziegler topology.The algebraic and model-theoretic relevance of the Ziegler topology is discussed in [Z], [P] and in many subsequent papers, including [P1], [P2] and [P3], for instance. Here we are interested in the Ziegler spectrum ZgRG of a group ring RG, where R is a Dedekind domain of characteristic 0 (for example R could be the ring Z of integers) and G is a finite group. In particular we deal with the R-torsionfree points of ZgRG.The main motivation for this is the study of RG-lattices (i.e., finitely generated R-torsionfree RG-modules).

Let G be the direct sum of the noncyclic groupof order four and a cyclic groupwhoseorderisthe power pn of some prime p. We show that ℤ2G-lattices have a decidable theory when the cyclotomic polynomia (x) is irreducible modulo 2ℤ for every j ≤ n. More generally we discuss the decision problem for ℤ2G-lattices when G is a finite group whose Sylow 2-subgroups are isomorphic to the noncyclic group of order four.

We propose a definition of weak o-minimality for structures expanding a Boolean algebra. We study this notion, in particular we show that there exist weakly o-minimal non o-minimal examples in this setting.