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Model Theory and Modules
, 1988
"... The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic 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 ..."
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Cited by 97 (23 self)
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The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic 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 ModR, the full subcategory of finitely presented modules will be denoted modR, the
Topological and Geometric aspects of the Ziegler Spectrum
, 1998
"... 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 ..."
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Cited by 9 (6 self)
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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 modR 7. The GabrielZariski topology 8. The sheaf of locally definable scalars 1 The Ziegler spectrum 1.1 A reminder on purity and pureinjectives 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 Rlinear 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 ; ...
Krull–Gabriel dimension and the modeltheoretic complexity of the category of modules over group rings of finite groups
 J. LONDON MATH. SOC
"... We classify group rings of finite groups over a field F according to the modeltheoretic 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. ..."
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We classify group rings of finite groups over a field F according to the modeltheoretic 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.
HOW TO CONSTRUCT A ‘CONCRETE’ SUPERDECOMPOSABLE PUREINJECTIVE MODULE OVER A STRING ALGEBRA
"... Abstract. We construct an element in a direct product of finite dimensional modules over a string algebra such that the pureinjective envelope of this element is a superdecomposable module. A nonzero module M is said to be superdecomposable if M has no indecomposable direct summands. For an examp ..."
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Abstract. We construct an element in a direct product of finite dimensional modules over a string algebra such that the pureinjective envelope of this element is a superdecomposable module. A nonzero module M is said to be superdecomposable if M has no indecomposable direct summands. For an example, let R be the endomorphism ring of a countable dimensional vector space V, and let I be the ideal of R consisting of all endomorphisms with finite dimensional images. If R ′ = R/I and e is a projection on a subspace of V whose image and coimage are infinite dimensional, then R ′ ∼ = eR ′ ⊕ (1 − e)R ′ as a right module over itself and R ′ ∼ = eR ′ ∼ = (1 − e ′)R ′. Furthermore, every nontrivial decomposition of R ′ as a right module is of this form, therefore R ′ is superdecomposable. For more examples, let R = k〈X, Y 〉 be a free algebra over a field k, and let E = E(RR) be the injective envelope of R considered as a right module over itself. It is easily verified (see [11, Prop. 8.36]) that RR has no nonzero uniform submodules. Therefore the same is true for E, hence E is a superdecomposable injective module. More generally, this is a common feature of finite dimensional wild algebras, that they usually (conjecturally always) have a superdecomposable pureinjective module (see [11, Ch. 8] for a list of existing results). Here a module M over a finite dimensional algebra A is pureinjective if M is a direct summand of a direct product of finite dimensional Amodules. If A is a tame finite dimensional algebra over a field, it has been believed for a while (see [24, p. 38]) that every pureinjective Amodule has an indecomposable direct summand. But recently Puninski [19] showed that every nondomestic string algebra over a countable field has a superdecomposable pureinjective module (note that every string algebra is tame). The main
TOWARDS THE DECIDABILITY OF THE THEORY OF MODULES OVER FINITE COMMUTATIVE RINGS
"... 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 Rmodu ..."
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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 Rmodules is undecidable, and verify decidability of this theory for some classes of tame finite commutative rings. 1.