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- President of the Societat Catalana de MatemàtiquesAssociated Professor at the Department of Mathematics, Universitat ... morePresident of the Societat Catalana de MatemàtiquesAssociated Professor at the Department of Mathematics, Universitat Autònoma de Barcelona (UAB).Member of the Barcelona Graduate School of Mathematics (BGSMath)Coordinator of the PhD Program in Mathematics at UABedit
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Research Interests: Mathematics, Algebraic Geometry, Homotopy Theory, Pure Mathematics, E, and 3 moreD, F, and M-projective module
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Let R anf T be rings, and let U be a faithful R-T-bimodule. We show that the triple (R, U, T) has a Morita duality if and only if RR and TT are linearly compact, and there is a duality between the semisimple factors R/J(R) and T/J(T)... more
Let R anf T be rings, and let U be a faithful R-T-bimodule. We show that the triple (R, U, T) has a Morita duality if and only if RR and TT are linearly compact, and there is a duality between the semisimple factors R/J(R) and T/J(T) induced by the socle of U. The main tool to prove this result is proving that lattice anti-isomorphisms can be patched together provided the AB-5* condition is satisfied.
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Let R be a hereditary, indecomposable, left pure-semisimple ring. We show that R has finite representation type if and only if a certain finitely presented module is endofinite, namely, the tilting and cotilting module W studied in L.... more
Let R be a hereditary, indecomposable, left pure-semisimple ring. We show that R has finite representation type if and only if a certain finitely presented module is endofinite, namely, the tilting and cotilting module W studied in L. Angeleri Hügel (2007) [2]. We then apply the tilting and the cotilting functors to study the endomorphism ring of W and its Auslander–Reiten components. Finally, we transfer this information to the category of right R-modules.
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We construct a family of semiprimitive and non von Neumann regular rings satisfying that any right or left module is isomorphic to a quotient of its flat cover (in the sense of Enochs) by a small submodule. This answers in the negative a... more
We construct a family of semiprimitive and non von Neumann regular rings satisfying that any right or left module is isomorphic to a quotient of its flat cover (in the sense of Enochs) by a small submodule. This answers in the negative a question posed by A. Amini, B. Amini, M. Ershad, and H. Sharif in 2007.
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Recently, tilting and cotilting classes over commutative noetherian rings have been classified in arXiv:1203.0907. We proceed and, for each n-cotilting class C, construct an n-cotilting module inducing C by an iteration of injective... more
Recently, tilting and cotilting classes over commutative noetherian rings have been classified in arXiv:1203.0907. We proceed and, for each n-cotilting class C, construct an n-cotilting module inducing C by an iteration of injective precovers. A further refinement of the construction yields the unique minimal n-cotilting module inducing C. Finally, we consider localization: a cotilting module is called ample, if all of its localizations are cotilting. We prove that for each 1-cotilting class, there exists an ample cotilting module inducing it, but give an example of a 2-cotilting class which fails this property.
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Abstract. We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag–Leffler ones. A right R–module M is called Baer if Ext1 R (M, T) = 0 for all torsion... more
Abstract. We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag–Leffler ones. A right R–module M is called Baer if Ext1 R (M, T) = 0 for all torsion modules T, and M is Mittag–Leffler in case the canonical map M ⊗R�i∈I Qi →�i∈I (M ⊗R Qi) is injective where {Qi}i∈I are arbitrary left R–modules. We show that a module M is Baer iff M is p–filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag–Leffler modules. Since the fundamental work of Ringel [29], the study of infinite dimensional modules has ...
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We use the concept of dual Goldie dimension and a characterization of semi-local rings due to Camps and Dicks (1993) to find some classes of modules with semi-local endomorphism ring. We deduce that linearly compact modules have... more
We use the concept of dual Goldie dimension and a characterization of semi-local rings due to Camps and Dicks (1993) to find some classes of modules with semi-local endomorphism ring. We deduce that linearly compact modules have semi-local endomorphism ring, cancel from direct sums and satisfy the n th root uniqueness property. We also deduce that modules over commutative rings satisfying AB5* also cancel from direct sums and satisfy the n th root uniqueness property. Let R be an associative ring with 1 and let Af be a right unital i?-module. A finite set Ax, ... , An of proper submodules of M is said to be coindependent if for each /, 1 < i < n, A¡ + f\j:jii Aj = M, and a family of submodules of M is said to be coindependent if each of its finite subfamilies is coindependent. The module M is said to have finite dual Goldie dimension if every coindependent family of submodules of M is finite. It can be shown that, in this case, there is a maximal coindependent family of submod...
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In this paper we study almost Krull-Schmidt modules, that is, modules with only finitely many direct sum decompositions up to isomorphism. This is a finiteness con- dition on modules that has nothing to do with the other finiteness... more
In this paper we study almost Krull-Schmidt modules, that is, modules with only finitely many direct sum decompositions up to isomorphism. This is a finiteness con- dition on modules that has nothing to do with the other finiteness conditions usually considered in the mathematical literature, like being noetherian, or AB5⁄, or having finite Goldie dimension, or finite dual Goldie dimension, or finite Krull dimension. Then we compute bounds on the number and the lengths of the direct sum decompositions of modules.