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We prove a transfer theorem which, when combined with the Jaffard-Kaplansky-Ohm Theorem, allows results in model theory of modules over B\'ezout domains to be translated into results over Pr\"ufer domains via their value groups. Extending work of Puninski and Toffalori, we show that the extended positive cone of the value group of a Pr\"ufer domain has m-dimension if and only if its lattice of pp-$1$-formulae has breadth (equivalently width) and that these dimensions are equal. Further, we show that the existence of these dimensions is equivalent to the lattice of pp-$1$-formulae having m-dimension (and hence to its Ziegler spectrum having Cantor-Bendixson rank) and the non-existence of superdecomposable pure-injective modules. Finally, we give a best possible upper bound for the m-dimension of the pp-$1$-lattice of a Pr\"ufer domain in terms of the m-dimension of the extended positive cone of its value group and show that all ordinals which are not of the form $\lambda+1$ for $\lambda$ a limit ordinal occur as the m-dimension of the pp-$1$-lattice of a B\'ezout domain., Comment: 35 pages

In this paper we give elementary conditions completely characterising when the theory of modules of a Pr\"ufer domain is decidable. Using these results, we show that the theory of modules of the ring of integer valued polynomials is decidable.

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Pr\"ufer (in particular B\'ezout) domains whose localizations at maximal ideals have dense value groups. For B\'ezout domains, these conditions are also necessary., Comment: 29 pages

Let $R$ be a discrete valuation domain with field of fractions $Q$ and maximal ideal generated by $\pi$. Let $\Lambda$ be an $R$-order such that $Q\Lambda$ is a separable $Q$-algebra. Maranda showed that there exists $k\in\mathbb{N}$ such that for all $\Lambda$-lattices $L$ and $M$, if $L/L\pi^k\simeq M/M\pi^k$ then $L\simeq M$. Moreover, if $R$ is complete and $L$ is an indecomposable $\Lambda$-lattice, then $L/L\pi^k$ is also indecomposable. We extend Maranda's theorem to the class of $R$-reduced $R$-torsion-free pure-injective $\Lambda$-modules. As an application of this extension, we show that if $\Lambda$ is an order over a Dedekind domain $R$ with field of fractions $Q$ such that $Q\Lambda$ is separable then the lattice of open subsets of the $R$-torsion-free part of the right Ziegler spectrum of $\Lambda$ is isomorphic to the lattice of open subsets of the $R$-torsion-free part of the left Ziegler spectrum of $\Lambda$. Finally, with $k$ as in Maranda's theorem, we show that if $M$ is $R$-torsion-free and $H(M)$ is the pure-injective hull of $M$ then $H(M)/H(M)\pi^k$ is the pure-injective hull of $M/M\pi^k$. We use this result to give a characterisation of $R$-torsion-free pure-injective $\Lambda$-modules and describe the pure-injective hulls of certain $R$-torsion-free $\Lambda$-modules.

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.

In this paper we prove that the Ziegler spectra of all serial rings are sober. We then use this proof to give a general framework for computing and understanding Ziegler spectra of uniserial rings up to topological indistinguishability. Finally, we illustrate this technique by computing the Ziegler spectra of all rank one uniserial domains up to topological indistinguishability., Comment: 30 pages

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Pr\"ufer (in particular B\'ezout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For B\'{e}zout domains these conditions are also necessary., Comment: Updated so that the title and abstract matches the published version. Other minor corrections and changes made

We describe the modules in the Ziegler closure of ray and coray tubes in module categories over finite-dimensional algebras. We improve slightly on Krause's result for stable tubes by showing that the inverse limit along a coray in a ray or coray tube is indecomposable, so in particular, the inverse limit along a coray in a stable tube is indecomposable. In order to do all this, we first describe the finitely presented modules over and the Ziegler spectra of iterated one-point extensions of valuation domains. Finally we give a sufficient condition for the $k$-dual of a $\Sigma$-pure-injective module over a $k$-algebra to be indecomposable.

We show that the common theory of all modules over a tubular algebra (over a recursive algebraically closed field) is decidable. This result supports a long standing conjecture of Mike Prest which says that a finite-dimensional algebra (over a recursively given field) is tame if and only its common theory of modules is decidable. Moreover, as a corollary, we are able to confirm this conjecture for the class of concealed canonical algebras over algebraically closed fields. These are the first examples of non-domestic algebras which have been shown to have decidable theory of modules., Comment: The results have been extended to cover all tubular algebras rather than just canonical algebras of tubular type - the title and abstract have been changed to reflect this. Moreover, the counter example to corollary 8.8 of Prest and Harland's paper "Modules with irrational slope over tubular algebras" has be replaced by a proof that that corollary does not hold for any tubular algebra. 47 pages

We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: that a representation embedding from ${\rm Mod}\mbox{-}S$ to ${\rm Mod}\mbox{-}R$ admits an inverse interpretation functor from its image and hence that, in this case, ${\rm Mod}\mbox{-}R$ interprets ${\rm Mod}\mbox{-}S$. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings. Finally we prove that if $R,S$ are finite dimensional algebras over an algebraically closed field and $I:{\rm Mod}\mbox{-}R\rightarrow{\rm Mod}\mbox{-}S$ is an interpretation functor such that the smallest definable subcategory containing the image of $I$ is the whole of ${\rm Mod}\mbox{-}S$ then, if $R$ is tame, so is $S$ and similarly, if $R$ is domestic, then $S$ also is domestic., Comment: More results added