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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.
Reflective subcategories
"... Reflective subcategories Given a full subcategory 3 of a category A, the existence of left 3approximations (or 3preenvelopes) completing diagrams in a unique way is equivalent to the fact that 3 is reflective in A, in the classical terminology of category theory. In the first part of the paper we ..."
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Reflective subcategories Given a full subcategory 3 of a category A, the existence of left 3approximations (or 3preenvelopes) completing diagrams in a unique way is equivalent to the fact that 3 is reflective in A, in the classical terminology of category theory. In the first part of the paper we establish, for a rather general A, the relationship between reflectivity and covariant finiteness of 3 in A, and generalize F'reyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflections with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories. In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized KrullSchmidt subcategories of ModR, and has some connections with the "puresemisimple conjecture".
LARGE TILTING MODULES AND REPRESENTATION TYPE
, 804
"... Abstract. We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L ..."
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Abstract. We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsionfree modules proven in [5] for tame hereditary algebras.
The Functor Category
 CATEGORICAL METHODS IN REPRESENTATION THEORY, BRISTOL, SEPT. 2012
, 2012
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THE TORSIONFREE PART OF THE ZIEGLER SPECTRUM OF THE KLEIN FOUR GROUP.
"... Abstract. We will describe the torsionfree 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. 1. ..."
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Abstract. We will describe the torsionfree 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. 1.