Pascual Jara
University of Granada, Álgebra, Faculty Member
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In the fuzzy theory of sets and groups, the use of α-levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong α-levels, it is possible to establish a one to one correspondence which makes possible... more
In the fuzzy theory of sets and groups, the use of α-levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong α-levels, it is possible to establish a one to one correspondence which makes possible doubly, a gradual and a functorial treatment of the fuzzy theory. The main result of this paper is to identify the class of fuzzy sets, respectively, fuzzy groups, with subcategories of the functorial categories Set (0, 1], resp., Gr (0, 1]. In this line, the algebraic potential of this theory will be reached, in forthcoming papers.
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The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules... more
The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules inside a generalized category of modules, in fact, a functor category, Mod−P, which is a Grothendieck category. To do that, first we consider the preadditive category P, defined by the interval P=(0,1], to build a torsionfree class J in Mod−P, and a hereditary torsion theory in Mod−P, to finally identify equivalence classes of fuzzy submodules of a module M with F-pair, which are pair (G,F), of decreasing gradual submodules of M, where G belongs to J, satisfying G=Fd, and ∪αF(α) is a disjoint union of F(1) and F(α)\G(α), where α is running in (0,1].
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This piece of work presents a simple and compact overview of the design problem of tensegrity structures in two and three dimensions. The main aim of this study is to present the design and calculation of tensegrity structures in their... more
This piece of work presents a simple and compact overview of the design problem of tensegrity structures in two and three dimensions. The main aim of this study is to present the design and calculation of tensegrity structures in their simplest form, avoiding unnecessary simplifications that can rule out solutions, as has happened up until now. As a result of the simplicity of the procedure, two types of tensegrity structures are obtained for the same initial topology: full and folded forms. Several examples are shown.
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In fuzzy theory of sets and groups, the use of $\alpha$--levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong $\alpha$--levels, it is possible to establish a one to one correspondence which makes... more
In fuzzy theory of sets and groups, the use of $\alpha$--levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong $\alpha$--levels, it is possible to establish a one to one correspondence which makes possible doubly, a gradual and a functorial treatment of the fuzzy theory. The main result of this paper is to identify the class of fuzzy sets, respectively fuzzy groups, with subcategories of the functorial categories $\mathcal{S}\textit{et}^{(0,1]}$, resp. $\mathcal{G}\textit{r}^{(0,1]}$.
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Ь б г и з лгж з иг з гл гл знб га гбдйи и гв в йз иг д ж гжб бйаи к ж и Ф ж в И Р жб и в ж г ви ждга и гв в ад йз иг й а бгж ж а зи ви ждга и в йв и гвзК и ж и гж и а вижг й и гв в л л в ано и гбда м ин г и б и г л з аа г йз гйж ии ви гв... more
Ь б г и з лгж з иг з гл гл знб га гбдйи и гв в йз иг д ж гжб бйаи к ж и Ф ж в И Р жб и в ж г ви ждга и гв в ад йз иг й а бгж ж а зи ви ждга и в йв и гвзК и ж и гж и а вижг й и гв в л л в ано и гбда м ин г и б и г л з аа г йз гйж ии ви гв гв дда и гвзК ... Хйаи к ж и ви ждга и гв гвз зиз в к в ...
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Abstract. A study of finiteness in a kind of finitely presented quotient algebras is displayed in this paper. The relations generating the ideals are given by monomials or binomials of same length, in order to obtain homogeneous... more
Abstract. A study of finiteness in a kind of finitely presented quotient algebras is displayed in this paper. The relations generating the ideals are given by monomials or binomials of same length, in order to obtain homogeneous computations. These ideals are parametrized by 3-tuples (a, b, c), being a the number of variables, b the length of monomials and c the number of relations conforming the ideal. We focus on the analysis of the (2, 3, 4)-family, constituted by 58905 elements, compute all the possible ideals, obtain the corresponding Gröbner- ...
Page 1. Algebras and Representation Theory (2005) 8: 363374 © Springer 2005 DOI: 10.1007/s00000-005-8110-3 Hereditary and Formally Smooth Coalgebras P. JARA1, L. MERINO1, D. LLENA2 and D. STEFAN3,⋆ 1Department ...
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In this note we propose an effective method based on the computation of a Gröbner basis of a left ideal to calculate the Gelfand-Kirillov dimension of modules.
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We develop the notion of primeness in coalgebras (over a commutative field). In particular, in this work we focus our attention on the study and characterization of prime subcoalgebras of path coalgebras of quivers and, by extension, of... more
We develop the notion of primeness in coalgebras (over a commutative field). In particular, in this work we focus our attention on the study and characterization of prime subcoalgebras of path coalgebras of quivers and, by extension, of prime pointed coalgebras.
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We establish the so-called Invariance under Twisting Theorems, stating that twisted tensor products built up from different algebras that relate in certain ways are canonically isomorphic. These results generalize (and provide categorical... more
We establish the so-called Invariance under Twisting Theorems, stating that twisted tensor products built up from different algebras that relate in certain ways are canonically isomorphic. These results generalize (and provide categorical versions of) several well-known results in Hopf algebra theory, such as the invariance of the smash product under Drinfeld twisting, or the isomorphism between the Drinfeld double and a certain smash product for quasitriangular Hopf algebras.
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Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H acts and coacts in a compatible way. To every modular crossed H-module M we associate a cyclic object Z(H,M). The cyclic homology of Z(H,M)... more
Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H acts and coacts in a compatible way. To every modular crossed H-module M we associate a cyclic object Z(H,M). The cyclic homology of Z(H,M) extends the usual cyclic homology of the algebra structure of H, and the relative cyclic homology of an H-Galois extension. For a Hopf subalgebra K we compute, under some assumptions, the cyclic homology of an induced modular crossed module. As a direct application of this computation, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the (usual) cyclic homology of group algebras and quantum tori. Finally, when H is the enveloping algebra of a Lie algebra, we construct a spectral sequence that converges to the cyclic homology of H with coefficients in an arbitrary modular crossed module. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of H with coe...
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For any commutative ring $A$ we introduce a generalization of $S$-noetherian rings using a hereditary torsion theory $\sigma$ instead of a multiplicatively closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally... more
For any commutative ring $A$ we introduce a generalization of $S$-noetherian rings using a hereditary torsion theory $\sigma$ instead of a multiplicatively closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $\sigma$-noetherian ring, then $\sigma$ is of finite type, and that totally $\sigma$-noetherian is a local property.
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ABSTRACT
The notion of uniform (or “topologizing”) filter has been introduced by P. Gabriel [Bull. Soc. Math. Fr. 90, 323-448 (1962; Zbl 0201.35602)], who proved that idempotent uniform filters (nowadays referred to as “Gabriel filters”) over a... more
The notion of uniform (or “topologizing”) filter has been introduced by P. Gabriel [Bull. Soc. Math. Fr. 90, 323-448 (1962; Zbl 0201.35602)], who proved that idempotent uniform filters (nowadays referred to as “Gabriel filters”) over a ring R correspond bijectively to localizations of the category R-mod. At the later stage, O. Goldman [J. Algebra 13, 10-47 (1969; Zbl 0201.04002)] has pointed out that Gabriel filters are also in bijective correspondence with idempotent kernel functors. In the past, uniform filters have mainly been considered within the framework of linear topologies. Recently, however, new applications of uniform filters arose in the context of noncommutative algebraic geometry. These applications require a deeper study of the functorial properties of uniform filters with respect to change of base ring and thus urged us to reconsider the notion of uniform filter. In the first section of the paper under review, the authors recollect some general results on the lattice...
MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1. K. Yano, Integral Formulas in Riemannian Geometry (1970) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) 3. VS Vladimirov, Equations of Mathematical Physics... more
MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1. K. Yano, Integral Formulas in Riemannian Geometry (1970) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) 3. VS Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. ...
We prove a characterization of a P$\star$MD, when $\star$ is a semistar operation, in terms of polynomials (by using the classical characterization of Pr\"{u}fer domains, in terms of polynomials given by R. Gilmer and J. Hoffman... more
We prove a characterization of a P$\star$MD, when $\star$ is a semistar operation, in terms of polynomials (by using the classical characterization of Pr\"{u}fer domains, in terms of polynomials given by R. Gilmer and J. Hoffman \cite{Gilmer/Hoffmann:1974}, as a model), extending a result proved in the star case by E. Houston, S.J. Malik and J. Mott \cite{Houston/Malik/Mott:1984}. We also deal
ABSTRACT Let A be a commutative ring. For any set P of prime ideals of A, we define a new ring Na(A,P): the Nagata ring. This new ring has the particularity that we may transform certain properties relative to P to properties on the whole... more
ABSTRACT Let A be a commutative ring. For any set P of prime ideals of A, we define a new ring Na(A,P): the Nagata ring. This new ring has the particularity that we may transform certain properties relative to P to properties on the whole ring Na(A,P); some of these properties are: ascending chain condition, Krull dimension, Cohen-Macaulay, Gorenstein. Our main aim is to show that most of the above properties relative to a set of prime ideals P (i.e., local properties) determine and are determined by the same properties on the Nagata ring (i.e., global properties). In order to look for new applications, we show that this construction is functorial, and exhibits a functorial embedding from the localized category (A,P)-Mod into the module category Na(A,P)-Mod.
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A GENERALIZATION OF SEMISIMPLE MODULES Jose L. Bueso and Pascual Jara Departamento de Algebra. Universidad de Granada. Espana. ABSTRACT. This communication generalizes the concepts of the socle of a module and of the semisimple module, by... more
A GENERALIZATION OF SEMISIMPLE MODULES Jose L. Bueso and Pascual Jara Departamento de Algebra. Universidad de Granada. Espana. ABSTRACT. This communication generalizes the concepts of the socle of a module and of the semisimple module, by replacing simples ...
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Resumen The papers in this proceedings volume are selected research papers in different areas of ring theory, including graded rings, differential operator rings, K-theory of noetherian rings, torsion theory, regular rings, cohomology of... more
Resumen The papers in this proceedings volume are selected research papers in different areas of ring theory, including graded rings, differential operator rings, K-theory of noetherian rings, torsion theory, regular rings, cohomology of algebras, local cohomology ...
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ABSTRACT It is well known that in some cases the functorExtRμ(−, R) defines a duality between module categories. In earlier papers we studied when this duality can be represented by a bimodule and have characterized when this happens. In... more
ABSTRACT It is well known that in some cases the functorExtRμ(−, R) defines a duality between module categories. In earlier papers we studied when this duality can be represented by a bimodule and have characterized when this happens. In this paper, using some computational methods of noncommutative Gröbner bases in the construction of projective resolutions of irreducible finite-dimensional representations, we show new examples of algebras satisfying this property.
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ABSTRACT In this paper we study relative duality theory, with respect to an idempotent kernel functor σ over some commutative ring R and prove that σ-dualizing R-modules are not only locally injective, but (somewhat surprisingly) globally... more
ABSTRACT In this paper we study relative duality theory, with respect to an idempotent kernel functor σ over some commutative ring R and prove that σ-dualizing R-modules are not only locally injective, but (somewhat surprisingly) globally injective. Using a relative version of completion, we show that the endomorphism ring of a σ-dualizing module coincides with the completion of R with respect to σ. In the final part of the paper we consider relative Gorenstein rings, giving an explicit calculation of their generalized local cohomology groups.
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In this paper we relate the completion introduced in [Bueso et al.(1994)] with the double dual, proving that if M is a σ-finitely generated R-module, then both notions coincide. In the case of a complete ring we prove a structure theorem... more
In this paper we relate the completion introduced in [Bueso et al.(1994)] with the double dual, proving that if M is a σ-finitely generated R-module, then both notions coincide. In the case of a complete ring we prove a structure theorem for reflexive modules and as a ...
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Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems... more
Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer, P. Lorenzen and P. Jaffard (cf. Halter–Koch's book [32]), in this paper we outline a general approach to the theory of Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is a semistar operation. This approach leads to relax the classical restriction on the base domain, which is not necessarily integrally closed in the semistar case, and to determine a semistar invariant character for this important class of multiplicative domains (cf. also J. M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J. B...
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Abstract In this work we continue studying the notion of completion ofR-modules, over a commutative ringR, relative to a torsion theoryϑ. We develop some techniques relative to localization at prime ideals and give structural results on... more
Abstract In this work we continue studying the notion of completion ofR-modules, over a commutative ringR, relative to a torsion theoryϑ. We develop some techniques relative to localization at prime ideals and give structural results on the completion of finitely ...