132 results on '"Torsion theory"'
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2. THE SIGNIFICANCE OF THE CONTRIBUTIONS OF CONGRUENCES TO THE THEORY OF CONNECTEDNESSES AND DISCONNECTEDNESSES FOR TOPOLOGICAL SPACES AND GRAPHS.
- Author
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VELDSMAN, STEFAN
- Abstract
This is a survey of some of the consequences of the recently introduced congruences on the theory of connectednesses (radical classes) and disconnectednesses (semisimple classes) of graphs and topological spaces. In particular, it is shown that the connectednesses and disconnectednesses can be obtained as Hoehnke radicals and a connectedness has a characterization in terms of congruences resembling the classical characterization of its algebraic counterpart using ideals for a radical class. But this approach has also shown that there are some unexpected differences and surprises: an ideal-hereditary Hoehnke radical of topological spaces or graphs need not be a Kurosh-Amitsur radical and in the category of graphs with no loops, non-trivial connectednesses and disconnectednesses exist, but all Hoehnke radicals degenerate. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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3. Some aspects about the lattice [τg,χ] and the torsion theories of Kaplansky-type.
- Author
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Pérez, Jaime Castro, Montes, José Ríos, and Sánchez, Gustavo Tapia
- Subjects
GENERALIZATION ,ATOMS - Abstract
In this manuscript, we define when a torsion theory, which is a generalization of Goldie's torsion theory, is of the type I, II, or III, according to Kaplansky's theory of types, and we establish some structure theorems of regular, right self-injective rings using the torsion theories of Kaplansky-type. We study the properties that each of these torsion theories has and their influence on the lattice [ τ g , χ ] of generalizations of Goldie's torsion theory, locating several subintervals of [ τ g , χ ] which contain no atoms nor coatoms, thus extending our knowledge of the lattice structure of [ τ g , χ ] . [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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4. Killing Weights from the Perspective of -Structures.
- Author
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Bondarko, Mikhail V. and Vostokov, Sergei V.
- Abstract
This paper is devoted to morphisms killing weights in a range (as defined by the first author) and to objects without these weights (as essentially defined by J. Wildeshaus) in a triangulated category endowed with a weight structure . We describe several new criteria for morphisms and objects to be of these types. In some of them we use virtual -truncations and a -structure adjacent to . In the case where the latter exists, we prove that a morphism kills weights if and only if it factors through an object without these weights; we also construct new families of torsion theories and projective and injective classes. As a consequence, we obtain some "weakly functorial decompositions" of spectra (in the stable homotopy category ) and a new description of those morphisms that act trivially on the singular cohomology with coefficients in an arbitrary abelian group . [ABSTRACT FROM AUTHOR]
- Published
- 2023
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5. Torsion theories connected by a heart and hearts of intervals.
- Author
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Yoshizawa, Takeshi
- Abstract
Intervals and hearts of intervals in the lattice of torsion parts for an abelian category are closely related to various concepts such as wide subcategory, ICE-closed subcategory, and tilting theory. On the other hand, in order to characterize the Melkersson condition suitable for local cohomology theory, we introduced the concept of torsion theory connected by a Serre subcategory in the category of modules. The paper shows that these seemingly unrelated concepts occur simultaneously in the larger context called the torsion theory connected by a heart. [ABSTRACT FROM AUTHOR]
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- 2023
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6. Torsion Theories and Coverings of V-Groups.
- Author
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Michel, Aline
- Abstract
For a commutative, unital and integral quantale V, we generalize to V-groups the results developed by Gran and Michel for preordered groups. We first of all show that, in the category V- Grp of V-groups, there exists a torsion theory whose torsion and torsion-free subcategories are given by those of indiscrete and separated V-groups, respectively. It turns out that this torsion theory induces a monotone-light factorization system that we characterize, and it is then possible to describe the coverings in V- Grp . We next classify these coverings as internal actions of a Galois groupoid. Finally, we observe that the subcategory of separated V-groups is also a torsion-free subcategory for a pretorsion theory whose torsion subcategory is the one of symmetric V-groups. As recently proved by Clementino and Montoli, this latter category is actually not only coreflective, as it is the case for any torsion subcategory, but also reflective. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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7. The realization problem for generalized torsion theories.
- Author
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Yoshizawa, Takeshi
- Subjects
TORSION - Abstract
A generalized torsion theory associated with a Serre subcategory was introduced to characterize the Melkersson condition. However, little is known about when generalized torsion theories can be realized using specialization closed subsets of the spectrum of a ring. In this article, we investigate the characterization of generalized torsion theories realized by such a subset. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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8. Homotopy torsion theories.
- Author
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Mantovani, Sandra, Messora, Mariano, and Vitale, Enrico M.
- Subjects
- *
HOMOTOPY theory , *ORTHOGONAL systems , *TORSION - Abstract
In the context of categories equipped with a structure of nullhomotopies, we introduce the notion of homotopy torsion theory. As special cases, we recover pretorsion theories as well as torsion theories in multi-pointed categories and in pre-pointed categories. Using the structure of nullhomotopies induced by the canonical string of adjunctions between a category A and the category Arr (A) of arrows, we give a new proof of the correspondence between orthogonal factorization systems in A and homotopy torsion theories in Arr (A) , avoiding the request on the existence of pullbacks and pushouts in A. Moreover, such a correspondence is extended to weakly orthogonal factorization systems and weak homotopy torsion theories. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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9. Relation between balanced pairs and TTF triples.
- Author
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Li, Weiqing
- Subjects
- *
NOETHERIAN rings , *LOCAL rings (Algebra) , *COMMUTATIVE rings , *BIJECTIONS , *OPEN-ended questions - Abstract
Let A be a complete and cocomplete abelian category with the additional assumption that any direct sum and product of short exact sequences are exact. We explore the relation between balanced pairs and TTF triples in A. The main results are: (1) every balanced pair in A induces a TTF triple; (2) if A has projective covers and injective envelopes, then every TTF triple in A gives rise to a balanced pair, and hence there is a bijection between the equivalence classes of balanced pairs and TTF triples in A ; (3) a balanced pair in A is quasi admissible if and only if its induced TTF triple is centrally splitting. Our first application of these results provide abundant rings over which every balanced pair is quasi admissible, including local rings, commutative semiperfect rings, and commutative Noetherian rings. Another application is the classification of equivalence classes of cohereditary balanced pairs over arbitrary rings. We also present counterexamples to [15, Open questions 3 and 5]. Finally, we prove that the answers to [15, Open questions 2 and 4] are positive for coherent rings with weak global dimension at most one. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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10. On perfectly generated weight structures and adjacent t-structures.
- Author
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Bondarko, Mikhail V.
- Abstract
This paper is dedicated to the study of smashing weight structures (these are the weight structures "coherent with coproducts"), and the application of their properties to t-structures. In particular, we prove that the hearts of compactly generated t-structures are Grothendieck abelian categories; this statement strengthens earlier results of several other authors. The central theorem of the paper is as follows: any perfect (as defined by Neeman) set of objects of a triangulated category generates a weight structure; we say that weight structures obtained this way are perfectly generated. An important family of perfectly generated weight structures are (the opposites to) the ones right adjacent to compactly generated t-structures; they give injective cogenerators for the hearts of the latter. Moreover, we establish the following not so explicit result: any smashing weight structure on a well generated triangulated category (this is a generalization of the notion of a compactly generated category that was also defined by Neeman) is perfectly generated; actually, we prove more than that. Furthermore, we give a classification of compactly generated torsion theories (these generalize both weight structures and t-structures) that extends the corresponding result of D. Pospisil and J. Šťovíček to arbitrary smashing triangulated categories. This gives a generalization of a t-structure statement due to B. Keller and P. Nicolas. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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11. A PRETORSION THEORY FOR THE CATEGORY OF ALL CATEGORIES.
- Author
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Xarez, João J.
- Subjects
TORSION ,FUNCTOR theory ,BLOWING up (Algebraic geometry) ,SYMMETRIC functions ,ALGEBRAIC equations - Abstract
Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2022
12. Localization and colocalization in tilting torsion theory for coalgebras.
- Author
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Li, Yuan and Yao, Hailou
- Abstract
Tilting theory plays an important role in the representation theory of coalgebras. This paper seeks how to apply the theory of localization and colocalization to tilting torsion theory in the category of comodules. In order to better understand the process, we give the (co)localization for morphisms, (pre)covers and special precovers. For that reason, we investigate the (co)localization in tilting torsion theory for coalgebras. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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13. From torsion theories to closure operators and factorization systems
- Author
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Marco Grandis and George Janelidze
- Subjects
exact sequence ,torsion theory ,closure operator ,factorization system ,ideal of null morphisms ,Mathematics ,QA1-939 - Abstract
Torsion theories are here extended to categories equipped with an ideal of 'null morphisms', or equivalently a full subcategory of 'null objects'. Instances of this extension include closure operators viewed as generalised torsion theories in a 'category of pairs', and factorization systems viewed as torsion theories in a category of morphisms. The first point has essentially been treated in [15].
- Published
- 2020
14. Shear modulus of single wood pulp fibers from torsion tests.
- Author
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Dauer, M., Wolfbauer, A., Seidlhofer, T., and Hirn, U.
- Subjects
MODULUS of rigidity ,WOOD-pulp ,FIBERS ,SULFATE pulping process ,ROCK groups - Abstract
The shear modulus of pulp fibers is difficult to measure and only very little literature is available on this topic. In this work we are introducing a method to measure this fiber property utilizing a custom built instrument. From the geometry of the fiber cross section, the fiber twisting angle and the applied torque, the shear modulus is derived by de Saint Venant's theory of torsion. The deformation of the fiber is applied by a moving coil mechanism. The support of the rotating part consists of taut bands, making it nearly frictionless, which allows easy control of the torque to twist the fiber. A permanent magnet moving coil meter was fitted with a sample holder for fibers and torque references. Measurements on fine metal bands were performed to validate the instrument. The irregular shape of the fibers was reconstructed from several microtome cuts and an apparent torsion constant was computed by applying de Saint Venant's torsion theory. Fibers from two types of industrial pulp were measured: thermomechanical pulp (TMP) and Kraft pulp. The average shear modulus was determined as (2.13 ± 0.36) GPa for TMP and (2.51 ± 0.50) GPa for Kraft fibers, respectively. The TMP fibers showed a smaller shear modulus but, due to their less collapsed state, a higher torsional rigidity than the kraft fibers. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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15. A GENERALIZATION OF PURELY EXTENDING MODULES RELATIVE TO A TORSION THEORY.
- Author
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DOĞRUŐZ, Semra and TARHAN, Azime
- Subjects
- *
GENERALIZATION , *CLASSIFICATION - Abstract
In this work we introduce a new concept, namely, purely τ s-extending modules (rings) which is torsion-theoretic analogues of extending modules and then we extend many results from extending modules to this new concept. For instance, we show that for any ring R with unit, RR is purely τ s-extending if and only if every cyclic τ-nonsingular R-module is flat. Also, we make a classification for the direct sums of the rings to be purely τ s-extending. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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16. Torsion theories and coverings of preordered groups.
- Author
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Gran, Marino and Michel, Aline
- Subjects
- *
ORDERED groups , *CATEGORIES (Mathematics) , *LINEAR orderings , *TORSION theory (Algebra) , *GALOIS theory , *FACTORIZATION - Abstract
We explore a non-abelian torsion theory in the category of preordered groups: the objects of its torsion-free subcategory are the partially ordered groups, whereas the objects of the torsion subcategory are groups (with the total order). The reflector from the category of preordered groups to this torsion-free subcategory has stable units, and we prove that it induces a monotone-light factorization system. We describe the coverings relative to the Galois structure naturally associated with this reflector, and explain how these coverings can be classified as internal actions of a Galois groupoid. Finally, we prove that in the category of preordered groups there is also a pretorsion theory, whose torsion subcategory can be identified with a category of internal groups. This latter is precisely the subcategory of protomodular objects in the category of preordered groups, as recently discovered by Clementino, Martins-Ferreira, and Montoli. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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17. A FRAMEWORK FOR TORSION THEORY COMPUTATIONS ON ELLIPTIC THREEFOLDS.
- Author
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ANGELES, DAVID, LO, JASON, and VAN DER LINDEN, COURTNEY M.
- Subjects
- *
TORSION theory (Algebra) , *SHEAF theory , *TOPOLOGY , *HOMOLOGICAL algebra - Abstract
We give a list of statements on the geometry of elliptic threefolds phrased only in the language of topology and homological algebra. Using only notions from topology and homological algebra, we recover existing results and prove new results on torsion pairs in the category of coherent sheaves on an elliptic threefold. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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18. Hereditary Torsion Theories for Graphs.
- Author
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Veldsman, S.
- Subjects
- *
GRAPH theory , *UNIVERSAL algebra , *UNDIRECTED graphs , *MATHEMATICAL connectedness , *TORSION theory (Algebra) - Abstract
Using congruences, a Hoehnke radical can be defined for graphs in the same way as for universal algebras. This leads in a natural way to the connectednesses and disconnectednesses (= radical and semisimple classes) of graphs. It thus makes sense to talk about ideal-hereditary Hoehnke radicals for graphs (= hereditary torsion theories). All such radicals for the category of undirected graphs which allow loops are explicitly determined. Moreover, in contrast to what is the case for the well-known algebraic categories, it is shown here that such radicals for graphs need not be Kurosh–Amitsur radicals. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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19. Pretorsion theories, stable category and preordered sets.
- Author
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Facchini, Alberto and Finocchiaro, Carmelo Antonio
- Abstract
We show that in the category of preordered sets there is a natural notion of pretorsion theory, in which the partially ordered sets are the torsion-free objects and the sets endowed with an equivalence relation are the torsion objects. Correspondingly, it is possible to construct a stable category factoring out the objects that are both torsion and torsion-free. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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20. The Torsion Theory and the Melkersson Condition.
- Author
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Yoshizawa, Takeshi
- Abstract
We consider a generalization of the notion of torsion theory, which is associated with a Serre subcategory over a commutative Noetherian ring. In 2008 Aghapournahr and Melkersson investigated the question of when local cohomology modules belong to a Serre subcategory of the module category. In their study, the notion of Melkersson condition was defined as a suitable condition in local cohomology theory. One of our purposes in this paper is to show how naturally the concept of Melkersson condition appears in the context of torsion theories. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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21. FI-extending hulls of Abelian groups.
- Author
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Birkenmeier, Gary F. and LeBlanc, Richard L.
- Abstract
All groups considered are Abelian. As is well known, in a divisible group every subgroup is essential in a direct summand. Moreover, the fully invariant subgroups play a crucial role in the structure of an Abelian group. Thus it is natural to consider the class of groups in which every fully invariant subgroup is essential in a direct summand. In this paper, we provide a construction for imbedding a torsion group into a FI-extending group in some "minimal" way. This imbedding preserves p-height. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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22. Injective stabilization of additive functors, I. Preliminaries.
- Author
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Martsinkovsky, Alex and Russell, Jeremy
- Subjects
- *
INJECTIVE functions , *STABILITY theory , *ADDITIVE functions , *FINITE fields , *INFINITY (Mathematics) - Abstract
This paper is the first one in a series of three dealing with the concept of injective stabilization of the tensor product and its applications. Its primary goal is to collect known facts and establish a basic operational calculus that will be used in the subsequent parts. This is done in greater generality than is necessary for the stated goal. Several results of independent interest are also established. They include, among other things, connections with satellites, an explicit construction of the stabilization of a finitely presented functor, various exactness properties of the injectively stable functors, a construction, from a functor and a short exact sequence, of a doubly-infinite exact sequence by splicing the injective stabilization of the functor and its derived functors. When specialized to the tensor product with a finitely presented module, the injective stabilization with coefficients in the ring is isomorphic to the 1-torsion functor. The Auslander-Reiten formula is extended to a more general formula, which holds for arbitrary (i.e., not necessarily finite) modules over arbitrary associative rings with identity. Weakening of the assumptions in the theorems of Eilenberg and Watts leads to characterizations of the requisite zeroth derived functors. The subsequent papers, provide applications of the developed techniques. Part II deals with new notions of torsion module and cotorsion module of a module. This is done for arbitrary modules over arbitrary rings. Part III introduces a new concept, called the asymptotic stabilization of the tensor product. The result is closely related to different variants of stable homology (these are generalizations of Tate homology to arbitrary rings). A comparison transformation from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be epic. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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23. A classification of torsion classes in abelian categories.
- Author
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Liu, Yong and Stanley, Donald
- Subjects
ABELIAN categories ,CLASSIFICATION ,TORSION theory (Algebra) - Abstract
We give a classification of torsion classes (or nullity classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda's classification of Serre subcategories. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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24. AN ORDER-THEORETIC PERSPECTIVE ON CATEGORIAL CLOSURE OPERATORS.
- Author
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ABDALLA, ABDURAHMAN MASOUD and JANELIDZE, ZURAB
- Subjects
CLOSURE operators ,LATTICE theory ,OPERATOR theory ,BANACH spaces ,PROBABILITY theory ,GRAPH theory ,TOPOLOGICAL spaces - Abstract
This paper deals with an order-theoretic analysis of certain structures studied in category theory. A categorical closure operator (cco in short) is a structure on a category, which mimics the structure on the category of topological spaces formed by closing subspaces of topological spaces. Such structures play a significant role not only in categorical topology, but also in topos theory and categorical algebra. In the case when the category is a poset, as a particular instance of the notion of a cco, one obtains what we call in this paper a binary closure operator (bco in short). We show in this paper that bco's allow one to see more easily the connections between standard conditions on general cco's, and furthermore, we show that these connections for cco's can be even deduced from the corresponding ones for bco's, when considering cco's relative to a well-behaved class of monorphisms as in the literature. The main advantage of the approach to such cco's via bco's is that the notion of a bco is self-dual (relative to the usual posetal duality), and by applying this duality to cco's, independent results on cco's are brought together. In particular, we can unify basic facts about hereditary closure operators with similar facts about minimal closure operators. Bco's also reveal some new links between categorical closure operators, the usual unary closure and interior operators, modularity law in order and lattice theory, the theory of factorization systems and torsion theory. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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25. Torsion classes generated by silting modules.
- Author
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Breaz, Simion and Žemlička, Jan
- Abstract
We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings, it is proved that these are exactly the torsion T such that the regular module has a special T -preenvelope. In particular, every torsion-enveloping class in Mod-R are of the form Gen(T) for a minimal silting module T. For the dual case, we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form Cogen(T), where T is a cosilting module. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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26. A Diagram of Galois Connections of Functorial Topologies.
- Author
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Castellini, Gabriele and Dziobiak, Stan
- Abstract
A Galois connection between functorial topologies on abelian groups and subclasses of abelian groups is constructed by means of the notion of indiscrete topology. It is shown that the composition of this Galois connection with a previously introduced one coincides with the classical Galois connection induced by the notion of constant morphism. Examples are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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27. New results on C and C lattices with applications to Grothendieck categories and torsion theories.
- Author
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Albu, Toma and Iosif, Mihai
- Subjects
- *
GROTHENDIECK categories , *TORSION theory (Algebra) , *MODULAR lattices , *MODULES (Algebra) , *ISOMORPHISM (Mathematics) - Abstract
In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions ( C) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions ( C) or ( C). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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28. A Torsion Theory in the Category of Cocommutative Hopf Algebras.
- Author
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Gran, Marino, Kadjo, Gabriel, and Vercruysse, Joost
- Abstract
The purpose of this article is to prove that the category of cocommutative Hopf K-algebras, over a field K of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it contains a torsion theory whose torsion-free and torsion parts are given by the category of groups and by the category of Lie K-algebras, respectively. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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29. Semi-localizations of semi-abelian categories.
- Author
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Gran, Marino and Lack, Stephen
- Subjects
- *
LOCALIZATION (Mathematics) , *ABELIAN functions , *MODULAR arithmetic , *MATHEMATICAL analysis , *RING theory - Abstract
A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. There are many interesting examples of semi-localizations, as for instance any torsion-free subcategory of a semi-abelian category. By specializing a result due to S. Mantovani, we first characterize the categories which are semi-localizations of exact Mal'tsev categories. We then prove a new characterization of protomodular categories in terms of binary relations, allowing us to obtain an abstract characterization of the semi-localizations of exact protomodular categories. This result is very useful to study the (hereditarily)-torsion-free subcategories of semi-abelian categories. Some examples are considered in detail in the categories of groups, crossed modules, commutative rings and topological groups. We finally explain how these results extend the corresponding ones obtained in the abelian context by W. Rump. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
30. t-Structures are Normal Torsion Theories.
- Author
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Fiorenza, Domenico and Loregiàn, Fosco
- Abstract
We characterize t-structures in stable ∞-categories as suitable quasicategorical factorization systems. More precisely we show that a t-structure 픱 on a stable ∞-category C is equivalent to a normal torsion theory 픽 on C, i.e. to a factorization system 픽 = (퓔, ℳ) where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
31. The conditions (Ci) in modular lattices, and applications.
- Author
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Albu, Toma, Iosif, Mihai, and Tercan, Adnan
- Subjects
- *
MODULAR lattices , *MODULES (Algebra) , *LATTICE theory , *CUBIC crystal system , *GROTHENDIECK categories , *TORSION theory (Algebra) - Abstract
In this paper, we introduce and investigate the latticial counterparts of the conditions (Ci), i = 1, 2, 3, 11, 12, for modules. In particular, we study the lattices satisfying the condition (C1), we call CC lattices (for Closed are Complements), i.e. the lattices such that any closed element is a complement, that are the latticial counterparts of CS modules (for Closed are Summands). Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
32. A Construction of Totally Reflexive Modules.
- Author
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Rahmati, Hamid, Striuli, Janet, and Wiegand, Roger
- Abstract
We construct infinite families of pairwise non-isomorphic indecomposable totally reflexive modules of high multiplicity. Under suitable conditions on the totally reflexive modules M and N, we find infinitely many non-isomorphic indecomposable modules arising as extensions of M by N. The construction uses the bimodule structure of ${Ext^{1}_{R}}((M,N)$ over the endomorphism rings of N and M. Our results compare with a recent theorem of Celikbas, Gheibi and Takahashi, and broaden the scope of that theorem. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
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33. Lattice preradicals with applications to Grothendieck categories and torsion theories.
- Author
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Albu, Toma and Iosif, Mihai
- Subjects
- *
LATTICE theory , *RADICAL theory , *GROTHENDIECK categories , *TORSION theory (Algebra) , *MODULES (Algebra) - Abstract
In this paper we introduce and investigate the latticial counterpart of the module-theoretical concept of preradical. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
34. s.Baer and s.Rickart Modules.
- Author
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Birkenmeier, Gary F. and LeBlanc, Richard L.
- Subjects
- *
MODULES (Algebra) , *RING theory , *SET theory , *TORSION theory (Algebra) , *NUMBER theory - Abstract
In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer modules satisfy a number of closure properties. Under certain conditions, a torsion theory is established for the s.Baer modules, and we provide examples of s.Baer torsion modules and modules with a nonzero s.Baer radical. The other principal interest of this paper is to provide explicit connections between s.Baer modules and projective modules. Among other results, we show that every s.Baer module is an essential extension of a projective module. Additionally, we prove, with limited and natural assumptions, that in a generalized triangular matrix ring every s.Baer submodule of the ring is projective. As an application, we show that every prime ring with a minimal right ideal has the strong summand intersection property. Numerous examples are provided to illustrate, motivate, and delimit the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
35. A RELATIVE EXTENDING MODULE AND TORSION PRECOVERS.
- Author
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BERKTAŞ, M. KEMAL and DOĞRUÖZ, S.
- Subjects
- *
MODULES (Algebra) , *TORSION theory (Algebra) , *HOMOMORPHISMS , *INTERSECTION theory , *ISOMORPHISM (Mathematics) - Abstract
We first characterize τ-complemented modules with relative (pre)covers. We also introduce an extending module relative to τ-pure submodules on a hereditary torsion theory τ and give its relationship with τ-complemented modules. [ABSTRACT FROM AUTHOR]
- Published
- 2015
36. MTame Modules And Local Gabriel Correspondence.
- Author
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Pérez, JaimeCastro and Montes, JoséRíos
- Subjects
VECTOR analysis ,MATHEMATICAL analysis ,VECTOR spaces ,FROBENIUS algebras ,ASSOCIATIVE algebras ,FROBENIUS groups ,GROUP theory - Abstract
Using the concept of prime submodule forM ∈ R-Mod,P ∈ Spec(M), andN ∈ σ[M], we define whenNisP-Mtame\ (Mtame) module. This concept generalizes the concept\ ofP-tame(tame) modules. ForM ∈ R-Mod and τ ∈M-tors, we use the concept of τM-Gabriel dimension and we study the relationship betweenMtamemodules and τM-Gabriel dimension. We find equivalent conditions for a moduleMprogenerator in σ[M] with τM-Gabriel dimension to have τM-Gabriel correspondence in terms of theP-Mtamemodules. This result extends the results by Albu et al. and Kim and Krause. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
37. Triangular matrix coalgebras and applications.
- Author
-
Iovanov, Miodrag Cristian
- Subjects
- *
MATRICES (Mathematics) , *ALGEBRA , *GENERALIZATION , *NOETHERIAN rings , *MODULES (Algebra) , *TORSION theory (Algebra) - Abstract
We formally introduce and study generalized comatrix coalgebras and upper triangular comatrix coalgebras, which are not only a dualization but also an extension of classical generalized matrix algebras. We use these to answer several open questions on Noetherian and Artinian type notions in the theory of coalgebras, and to give complete connections between these. We also solve completely the so called finite splitting problem for coalgebras: we show thatis a coalgebra such that the rational part of every left finitely generated-module splits off if and only ifis an upper triangular matrix coalgebra, for a serial coalgebrawhose Ext-quiver is a finite union of cycles, a finite dimensional coalgebraand a finite dimensional bicomodule. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
38. A Factorization Theorem for Topological Abelian Groups.
- Author
-
Dikranjan, Dikran and Bruno, AnnaGiordano
- Subjects
FACTORIZATION ,TOPOLOGICAL groups ,ABELIAN groups ,PONTRYAGIN duality ,TORSION theory (Algebra) ,DIVISIBILITY groups - Abstract
Using the nice properties of thew-divisible weight and thew-divisible groups, we prove a factorization theorem for compact abelian groupsK; namely,K = Ktor × Kd, whereKtoris a bounded torsion compact abelian group andKdis aw-divisible compact abelian group. By Pontryagin duality this result is equivalent to the same factorization for discrete abelian groups proved in [9]. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
39. Semiartinian Profinite Algebras have Nilpotent Jacobson Radical.
- Author
-
Iovanov, Miodrag
- Abstract
We give a method to study the finiteness of the coradical filtration of a coalgebra; as a consequence, we show that a left semiartinian profinite algebra has nilpotent Jacobson radical and is right semiartinian too. Equivalently, we show that a for a semilocal profinite algebra, T-nilpotence implies nilpotence for the Jacobson radical. This answers two open questions from Iovanov et al. (J Algebra 320(5):2144-2155, ). [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
40. Perverse Coherent t-Structures Through Torsion Theories.
- Author
-
Vitória, Jorge
- Abstract
Bezrukavnikov, later together with Arinkin, recovered Deligne's work defining perverse t-structures in the derived category of coherent sheaves on a projective scheme. We prove that these t-structures can be obtained through tilting with respect to torsion theories, as in the work of Happel, Reiten and Smalø. This approach allows us to define, in the quasi-coherent setting, similar perverse t-structures for certain noncommutative projective planes. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
41. Krull Dimension and Classical Krull Dimension of Modules.
- Author
-
Castro Pérez, Jaime and Montes, JoséRíos
- Subjects
KRULL rings ,MODULES (Algebra) ,DIMENSIONAL analysis ,PRIME numbers ,MATHEMATICAL bounds ,ABELIAN categories - Abstract
Using the concept of prime submodule defined by Raggi et al. in [16], forM ∈ R-Mod we define the concept of classical Krull dimension relative to a hereditary torsion theory τ ∈M-tors. We prove that ifMis progenerator in σ[M], τ ∈M-tors such thatMhas τ-Krull dimension thencl.Kτdim (M) ≤ kτ(M). Also we show that ifMis noetherian, τ-fully bounded, progenerator of σ[M], andM∈ 𝔽τ, thencl·Kτdim (M) = kτ(M). [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
42. Flat endofinite modules, prime ideals, and duality.
- Author
-
Rump, Wolfgang and Schmider, Simon
- Subjects
- *
FINITE fields , *MODULES (Algebra) , *PRIME ideals , *DUALITY theory (Mathematics) , *MATHEMATICAL functions - Abstract
Abstract: Bijective correspondences are established between endofinite injective left modules, endofinite flat right modules, finite collections of minimal noetherian prime ideals, normalized rank functions on left ideals and characters. Endofinite flat modules are identified as flat covers of modules associated to a minimal noetherian prime ideal, while endofinite flat injectives are characterized by localizations with a semiprimary QF-3 quotient ring. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
43. The Osofsky-Smith Theorem for Modular Lattices, and Applications (II).
- Author
-
Albu, Toma
- Subjects
MATHEMATICS theorems ,MODULES (Algebra) ,LATTICE theory ,GROTHENDIECK groups ,MATHEMATICAL analysis ,TORSION theory (Algebra) - Abstract
This is the second part of the paper with the same title published inCommunications in Algebrain 2011. It contains applications of the Latticial Osofsky–Smith Theorem to Grothendieck categories and module categories equipped with a torsion theory. Various many different meanings spread in the literature of the relative concepts with respect to a hereditary torsion theory τ on Mod-Rlike τ-essential submodule, τ-complement submodule, τ-CS module, etc. are also discussed. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
- Full Text
- View/download PDF
44. Torsion Pairs and Rigid Objects in Tubes.
- Author
-
Baur, Karin, Buan, Aslak, and Marsh, Robert
- Abstract
We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Prüfer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs, maximal rigid objects and the bijection between them geometrically. We also give a similar geometric description in the case of the linear orientation of a Dynkin quiver of type A. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
45. Tilting Modules Arising from Two-Term Tilting Complexes.
- Author
-
Abe, Hiroki
- Subjects
MODULES (Algebra) ,MATHEMATICAL complex analysis ,ARTIN algebras ,HOMOLOGY theory ,GROUP theory ,ENDOMORPHISMS - Abstract
We show that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine the endomorphism algebra of such a homology group, which is given as a certain factor algebra of the endomorphism algebra of the two-term tilting complex. Thus, every derived equivalence between Artin algebras given by a two-term tilting complex induces a derived equivalence between the corresponding factor algebras. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
- Full Text
- View/download PDF
46. Monotone-light factorisation systems and torsion theories.
- Author
-
Gran, Marino and Everaert, Tomas
- Subjects
- *
MONOTONE operators , *FACTORIZATION , *TORSION theory (Algebra) , *ABELIAN categories , *STABILITY theory , *TOPOLOGICAL groups - Abstract
Abstract: Given a torsion theory in an abelian category , the reflector to the torsion-free subcategory induces a reflective factorisation system on . It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Paré that induces a monotone-light factorisation system by simultaneously stabilising and localising , whenever the torsion theory is hereditary and any object in is a quotient of an object in . We extend this result to arbitrary normal categories, and improve it also in the abelian case, where the heredity assumption on the torsion theory turns out to be redundant. Several new examples of torsion theories where this result applies are then considered in the categories of abelian groups, groups, topological groups, commutative rings, and crossed modules. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
47. Grade of Ideals with Respect to Torsion Theories.
- Author
-
Asgharzadeh, Mohsen and Tousi, Massoud
- Subjects
IDEALS (Algebra) ,TORSION ,COMPARATIVE studies ,COMMUTATIVE rings ,RING theory ,NOETHERIAN rings ,MODULES (Algebra) ,COHOMOLOGY theory - Abstract
In this article we define and compare different types of the notion of grade with respect to torsion theories over commutative rings which are not necessarily Noetherian. We do this by using Ext-modules, Koszul cohomology modules, and Čech and local cohomology modules. An application of these results is given. [ABSTRACT FROM PUBLISHER]
- Published
- 2013
- Full Text
- View/download PDF
48. On Covers and Envelopes in Some Functor Categories.
- Author
-
Mao, Lixin
- Subjects
ENVELOPES (Geometry) ,FUNCTOR theory ,CATEGORIES (Mathematics) ,EXISTENCE theorems ,SPECIAL functions ,MODULES (Algebra) ,RING theory - Abstract
We study the existence of covers and envelopes by some special functors on the category of finitely presented modules. As an application, we characterize some important rings using these functors. We also investigate homological properties of some functors on the stable module category. The relationship between phantoms and Ext-phantoms is obtained. It is shown that every leftR-moduleMhas an Ext-phantom preenvelopef:M → Nwith coker(f) pure-projective. Finally, we prove that, as a torsionfree class of (mod-R, Ab), (mod-R, Ab) is generated by theFP-injective objects. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
49. Prime Submodules and Local Gabriel Correspondence in σ[ M ].
- Author
-
Pérez, JaimeCastro and Montes, JoséRíos
- Subjects
MODULES (Algebra) ,ISOMORPHISM (Mathematics) ,SET theory ,INDECOMPOSABLE modules ,TORSION theory (Algebra) ,ALGEBRA ,MATHEMATICAL analysis - Abstract
We consider the concept of prime submodule defined by Raggi et al. [7]. We find equivalent conditions for a module M progenerator in σ[M], with τ M -Gabriel dimension, to have a one-to-one correspondence between the set of isomorphism classes of indecomposable τ-torsion free injective modules in σ[M] and the set of τ-pure submodules prime in M, where τ is a hereditary torsion theory in σ[M]. Also we give a relation between the concept of prime M-ideal given by Beachy and the concept of prime submodule in M. We obtain that if M is progenerator in σ[M], then these concepts are equivalent. [ABSTRACT FROM PUBLISHER]
- Published
- 2012
- Full Text
- View/download PDF
50. Cotorsion Theories Cogenerated by a Torsionfree Class.
- Author
-
Fu, Xianhui, Zhu, Haiyan, and Sun, Mingyan
- Subjects
TORSION theory (Algebra) ,CATEGORIES (Mathematics) ,MODULES (Algebra) ,MATHEMATICAL proofs ,SET theory ,MATHEMATICAL analysis ,NUMERICAL analysis - Abstract
Let R be a right perfect ring, and let (ℱ, 𝒞) be a cotorsion theory in the category of right R-modules ℳ R . In this article, it is shown that every right R-module has a superfluous ℱ-cover if and only if there exists a torsion theory (𝒜, ℬ) such that (ℱ, 𝒞) is cogenerated by ℬ. It is also proved that if (𝒜, ℬ) is a cosplitting torsion theory, then (⊥ℬ, (⊥ℬ)⊥) is a hereditary and complete cotorsion theory, and if (𝒜, ℬ) is a centrally splitting torsion theory, then (⊥ℬ, (⊥ℬ)⊥) is a hereditary and perfect cotorsion theory. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
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