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 $ .

Furthermore, 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 characterization of R-torsion-free pure-injective $\Lambda $ -modules and describe the pure-injective hulls of certain R-torsion-free $\Lambda $ -modules.

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

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 describe the Ziegler spectrum of a Bézout domain B=D+XQ[X] where D is a principal ideal domain and Q is its field of fractions; in particular we compute the Cantor–Bendixson rank of this space. Using this, we prove the decidability of the theory of B-modules when D is “sufficiently” recursive.

The general theory of locally coherent Grothendieck categories is presented. To each locally coherent Grothendieck category $\C$ a topological space, the Ziegler spectrum of $\C,$ is associated. It is proved that the open subsets of the Ziegler spectrum of $\C$ are in bijective correspondence with the Serre subcategories of $\coh \C,$ the subcategory of coherent objects of $\C.$ This is a Nullstellensatz for locally coherent Grothendieck categories. If $R$ is a ring, there is a canonical locally coherent Grothendieck category $\RC$ (respectively, $\CR$) used for the study of left (respectively, right) $R$-modules. This category contains the category of $R$-modules and its Ziegler spectrum is quasi-compact, a property used to construct large (not finitely generated) indecomposable modules over an artin algebra. Two kinds of examples of locally coherent Grothendieck categories are given: the abstract category theoretic examples arising from torsion and localization and the examples that arise from particular modules over the ring $R.$ The duality between $\coh (\RC)$ and $\coh \CR$ is shown to give an isomorphism between the topologies of the left and right Ziegler spectra of a ring $R.$ The Nullstellensatz is used to give a proof of the result of Crawley-Boevey that every character $\xi: K_0 (\coh \C) \to Z$ is uniquely expressible as a $Z$-linear combination of irreducible characters.

1991 Mathematics Subject Classification: 16D90, 18E15.