The model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module ” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the

The aim here is to emphasise the topological and geometric structure that the Ziegler spectrum carries and to illustrate how this structure may be used in the analysis of particular examples. There is not space here for me to give a survey of what is known about the Ziegler spectrum so there are a number of topics that I will just mention in order to give some indication of what lies beyond what is discussed here. 1. The Ziegler spectrum 2. Various dimensions 3. These dimensions for artin algebras 4. These dimensions in general 5. Duality 6. The complexity of morphisms in mod-R 7. The Gabriel-Zariski topology 8. The sheaf of locally definable scalars 1 The Ziegler spectrum 1.1 A reminder on purity and pure-injectives Suppose that M is a submodule of N . Consider a finite system \Sigma n i=1 x i r ij = a j (j = 1; :::m) of R-linear equations over M : that is, the r ij are in R, the 1 a j are in M and the x i are indeterminates. Suppose that there is a solution b 1 ; ...

Abstract. 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 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. 1.

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Using the description of the Ziegler spectrum we characterise modules with various stability-theoretic properties (!-stability, superstability, categoricity) over certain classes of finite-dimensional algebras. We also

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