253 results on '"Torsion theory"'
Search Results
2. Localization and colocalization in tilting torsion theory for coalgebras
- Author
-
Yuan Li and Hailou Yao
- Subjects
Pure mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Tilting theory ,Colocalization ,Order (ring theory) ,01 natural sciences ,Representation theory ,Morphism ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Torsion theory ,0101 mathematics ,Mathematics - 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.
- Published
- 2021
3. Cotorsion torsion triples and the representation theory of filtered hierarchical clustering
- Author
-
Johan Steen, Steffen Oppermann, Magnus Bakke Botnan, Ulrich Bauer, and Mathematics
- Subjects
Pure mathematics ,General Mathematics ,Modulo ,01 natural sciences ,Representation theory ,Torsion theory ,Hierarchical clustering ,Quiver representation theory ,16G20, 16S90 (primary), 55N99 (secondary) ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Representation Theory (math.RT) ,0101 mathematics ,Cluster analysis ,Mathematics::Representation Theory ,Finite set ,Mathematics ,Subcategory ,Persistent homology ,010102 general mathematics ,Mathematics::Rings and Algebras ,Multiparameter persistence ,Torsion (algebra) ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
We give a full classification of representation types of the subcategories of representations of an $m \times n$ rectangular grid with monomorphisms (dually, epimorphisms) in one or both directions, which appear naturally in the context of clustering as two-parameter persistent homology in degree zero. We show that these subcategories are equivalent to the category of all representations of a smaller grid, modulo a finite number of indecomposables. This equivalence is constructed from a certain cotorsion torsion triple, which is obtained from a tilting subcategory generated by said indecomposables., Comment: 39 pages; corrected the lists appearing in Cor. 1.6 and minor changes throughout
- Published
- 2020
4. n-Coherence Relative to a Hereditary Torsion Theory
- Author
-
Zhu Zhanmin
- Subjects
Pure mathematics ,Ring (mathematics) ,Mathematics::Commutative Algebra ,Article Subject ,General Mathematics ,010102 general mathematics ,Coherence (statistics) ,01 natural sciences ,Integer ,Torsion theory ,QA1-939 ,0101 mathematics ,Mathematics - Abstract
Let R be a ring, τ=T,ℱ a hereditary torsion theory of mod-R, and n a positive integer. Then, R is called right τ-n-coherent if every n-presented right R-module is τ,n+1-presented. We present some characterizations of right τ-n-coherent rings, as corollaries, and some characterizations of right n-coherent rings and right τ-coherent rings are obtained.
- Published
- 2020
5. Kappa-Slender Modules
- Author
-
Radoslav Dimitric
- Subjects
General Mathematics ,k-tailwise slender ,16D10, 16D80, 16D90, 16N80, 16S10 03Exx, 03E55 ,infinite coproducts ,Physics::Fluid Dynamics ,Combinatorics ,k-coordinatewise slender ,filtered products ,QA1-939 ,FOS: Mathematics ,[MATH]Mathematics [math] ,Mathematics ,Physics::Biological Physics ,the hom functor ,torsion theory ,Mathematics - Rings and Algebras ,infinite products ,non-measurable cardinal ,Mathematics::Logic ,k-cslender ,secondary: 03c20, 03e10, 03e20, 03e55, 03e75, 20k25 ,Rings and Algebras (math.RA) ,primary: 16d80, 16d90, 18a20, 18a30, 18a40 ,slender module ,kappa-slender module ,k-tslender ,Kappa ,k-hmodule - Abstract
For an arbitrary infinite cardinal $\kappa$, we define classes of coordinatewise $\kappa$-slender and tailwise $\kappa$-slender modules as well as related classes of $h\kappa$-modules and initiate a study of these classes., Comment: A long unpublished note
- Published
- 2020
6. Assessment of ASCE 7–16 Seismic Isolation Bearing Torsional Displacement
- Author
-
Wael M. Hassan
- Subjects
021110 strategic, defence & security studies ,Damping ratio ,business.industry ,0211 other engineering and technologies ,Torsion (mechanics) ,020101 civil engineering ,02 engineering and technology ,Structural engineering ,0201 civil engineering ,Seismic hazard ,Torsion theory ,Seismic isolation ,Base isolation ,business ,Civil and Structural Engineering ,Parametric statistics ,Mathematics - Abstract
Base isolation provisions in ASCE 7 Standard have been historically shown to be conservative in estimating seismic demands. New torsional bearing base isolation displacement expressions have been proposed recently by the latest ASCE 7–16 Standard. This study compares the accuracy of the new ASCE 7–16 static-based isolation expressions for additional bearing displacement due to plane torsion to the response obtained using simplified structural dynamics’ plane torsion theory expressions. In addition, it conducts a parametric study to assess the effect of accidental eccentricity, damping ratio, and plan aspect ratio on the accuracy of ASCE 7–16 bearing displacement expression. The results showed that the ASCE 7–16 undamped displacement estimations improved compared to the significant conservatism of ASCE 7–10 by 7–33% depending on the eccentricity condition and ratio, which may further promote the use of base isolation in the US as a seismic hazard mitigation solution. However, the study also revealed that a considerable conservatism of the new base isolation bearing displacement provisions of ASCE 7–16 still exists in some cases, which ranged from 52 to 105% in the case of three equal fundamental frequencies and 5–20% in the case of equal lateral frequency and distinct torsional frequency. The study also showed that the ASCE 7–16 conservatism is proportional to the eccentricity and is more pronounced with biaxial eccentricity compared to single eccentricity. Furthermore, the results show that ASCE 7–16 expression accuracy significantly declines with damped systems with a possibility of un-conservative estimation of bearing displacements that can reach 25–40% with larger eccentricities. In addition, the ASCE 7–16 torsional displacement was shown to be inversely proportional to the plan aspect ratio with a discrepancy that can reach ± 20%. The study also exhibited that the ASCE 7–16 torsional displacement conservatism is slightly affected by increasing damping ratio above 4%.
- Published
- 2019
7. The torsion theory and the Melkersson condition
- Author
-
Takeshi Yoshizawa
- Subjects
Subcategory ,Noetherian ring ,Pure mathematics ,Mathematics::Commutative Algebra ,010102 general mathematics ,Local cohomology ,01 natural sciences ,Mathematics::K-Theory and Homology ,Ordinary differential equation ,Torsion theory ,Torsion (algebra) ,0101 mathematics ,Commutative property ,Mathematics - 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.
- Published
- 2019
8. Comultiplication modules relative to a hereditary torsion theory
- Author
-
Seçil Çeken
- Subjects
Pure mathematics ,Algebra and Number Theory ,Torsion theory ,Identity (philosophy) ,media_common.quotation_subject ,010102 general mathematics ,010103 numerical & computational mathematics ,Commutative ring ,0101 mathematics ,01 natural sciences ,media_common ,Mathematics - Abstract
Let R be a commutative ring with identity and τ be a hereditary torsion theory on R-Mod. In this article, we introduce and study the concept of τ-comultiplication module. We present several propert...
- Published
- 2019
9. A classification of torsion classes in abelian categories
- Author
-
Donald Stanley and Yong Liu
- Subjects
Pure mathematics ,Algebra and Number Theory ,Torsion theory ,010102 general mathematics ,Torsion (algebra) ,010103 numerical & computational mathematics ,Abelian category ,0101 mathematics ,Abelian group ,01 natural sciences ,Mathematics - 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 clas...
- Published
- 2018
10. An extension of S-artinian rings and modules to a hereditary torsion theory setting
- Author
-
Pascual Jara
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Generalization ,Multiplicative function ,Mathematics::Rings and Algebras ,Commutative ring ,Extension (predicate logic) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Torsion theory ,13E05, 13E10 ,FOS: Mathematics ,Mathematics - Abstract
For any commutative ring $A$ we introduce a generalization of $S$--artinian rings using a hereditary torsion theory $\sigma$ instead of a multiplicative closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $\sigma$--artinian ring, then $\sigma$ must be of finite type, and $A$ is totally $\sigma$--noetherian., Comment: 22 pages
- Published
- 2021
- Full Text
- View/download PDF
11. The lattice R-tors for perfect rings
- Author
-
Hugo Alberto Rincón-Mejía
- Subjects
Combinatorics ,Perfect ring ,Pure mathematics ,Class (set theory) ,Ring (mathematics) ,General Mathematics ,Torsion theory ,Lattice (group) ,Element (category theory) ,Mathematics - Abstract
We define ̃F in R-tors by r ̃F σ iff the class of r-codivisible modules coincides with the class of σ -codivisible modules. We prove that if R is left perfect ring (resp. semiperfect ring) then every [r] f Є R-tors/ ̃F (resp. [X]F and [ε]F) is a complete sublattice of R-tors We describe the largest element in [r] as X(Rad R/t,(Rad R)) and the least element of [r] as ε (t r(RadR)) Using these results we give a necessary and sufficient condition for the central splitting of Goldman torsion theory when R is semiperfect. We prove that for a QF ring R the least element of [X] ̃F is the Goldie torsion theory. This can be used to prove that for a QF ring ̃F and ̃T are equal, where r ̃T o iff the class of r-injective modules coincides with the class of σ-injective modules .
- Published
- 2021
12. Torsion theories and coverings of preordered groups
- Author
-
Marino Gran, Aline Michel, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
- Subjects
Pure mathematics ,Galois theory ,Structure (category theory) ,0102 computer and information sciences ,01 natural sciences ,Torsion theory ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,FOS: Mathematics ,Order (group theory) ,Category Theory (math.CT) ,Preordered groups ,0101 mathematics ,Algebra over a field ,Mathematics ,Subcategory ,Algebra and Number Theory ,010102 general mathematics ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Factorization system ,Rings and Algebras (math.RA) ,010201 computation theory & mathematics ,Torsion (algebra) ,18E40, 18E50, 06F15, 18A40 - Abstract
In this article 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., 21 pages, minor changes and improvements
- Published
- 2021
13. Pretorsion theories, stable category and preordered sets
- Author
-
Carmelo Antonio Finocchiaro and Alberto Facchini
- Subjects
Pure mathematics ,Applied Mathematics ,Stable category ,010102 general mathematics ,Preorder ,01 natural sciences ,Category of preordered sets ,Torsion theory ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,0103 physical sciences ,Torsion (algebra) ,Equivalence relation ,010307 mathematical physics ,0101 mathematics ,Partially ordered set ,Mathematics - 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.
- Published
- 2020
14. A canonical torsion theory for pro-p Iwahori–Hecke modules
- Author
-
Rachel Ollivier and Peter Schneider
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Iwahori subgroup ,Reductive group ,01 natural sciences ,Cohomology ,Residue field ,Torsion theory ,Mod ,0103 physical sciences ,Torsion (algebra) ,010307 mathematical physics ,Locally compact space ,0101 mathematics ,Mathematics - Abstract
Let F be a locally compact nonarchimedean field with residue characteristic p and G the group of F -rational points of a connected split reductive group over F . We define a torsion pair in the category Mod ( H ) of modules over the pro- p -Iwahori Hecke k -algebra H of G , where k is an arbitrary field. We prove that, under a certain hypothesis, the torsionfree class embeds fully faithfully into the category Mod I ( G ) of smooth k -representations of G generated by their pro- p -Iwahori fixed vectors. If the characteristic of k is different from p then this hypothesis is always satisfied and the torsionfree class is the whole category Mod ( H ) . If k contains the residue field of F then we study the case G = S L 2 ( F ) . We show that our hypothesis is satisfied, and we describe explicitly the torsionfree and the torsion classes. If F ≠ Q p and p ≠ 2 , then an H -module is in the torsion class if and only if it is a union of supersingular finite length submodules; it lies in the torsionfree class if and only if it does not contain any nonzero supersingular finite length module. If F = Q p , the torsionfree class is the whole category Mod ( H ) , and we give a new proof of the fact that Mod ( H ) is equivalent to Mod I ( G ) . These results are based on the computation of the H -module structure of certain natural cohomology spaces for the pro- p -Iwahori subgroup I of G .
- Published
- 2018
15. Relatively Lifting Modules.
- Author
-
Crivei, Septimiu
- Subjects
- *
ISOMORPHISM (Mathematics) , *GROUP theory , *HOMOMORPHISMS , *ABELIAN groups , *MATHEMATICS - Abstract
We consider a generalization of lifting modules relative to a class $\mathcal{A}$ of modules and a proper class 피 of short exact sequences of modules. These modules will be called 피-$\mathcal{A}$-lifting. We establish characterizations of modules with the property that every direct sum of copies of them is 피-$\mathcal{A}$-lifting. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
16. On a Generalization of Tilting Modules.
- Author
-
Zhang, Xiaoxiang and Yao, Lingling
- Subjects
MATHEMATICS ,MODULES (Algebra) ,ALGEBRA ,REASONING ,GENERALIZATION ,IDEALS (Algebra) - Abstract
Let R be a ring. A right R-module U is called Tor-tilting if [image omitted], where U+ = Hom(U, /), Cogen(U+) is the class of left R-modules cogenerated by U+ and [image omitted] consists of modules RM such that [image omitted]. Some examples and characterizations of Tor-tilting modules are given. Among others, it is shown that UR is Tor-tilting if and only if U+ is cotilting. Moreover, both tilting modules and completely faithful flat modules are proved to be Tor-tilting. The torsion theory induced by a Tor-tilting module is also investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
17. PERFECT SYMMETRIC RINGS OF QUOTIENTS.
- Author
-
VAŠ, LIA
- Subjects
- *
QUOTIENT rings , *RING theory , *MATHEMATICAL symmetry , *ASSOCIATIVE rings , *MATHEMATICS - Abstract
Perfect Gabriel filters of right ideals and their corresponding right rings of quotients have the desirable feature that every module of quotients is determined solely by the right ring of quotients. On the other hand, symmetric rings of quotients have a symmetry that mimics the commutative case. In this paper, we study rings of quotients that combine these two desirable properties. We define the symmetric versions of a right perfect ring of quotients and a right perfect Gabriel filter — the perfect symmetric ring of quotients and the perfect symmetric Gabriel filter and study their properties. Then we prove that the standard construction of the total right ring of quotients $Q^r_{\rm tot}(R)$ can be adapted to the construction of the largest perfect symmetric ring of quotients — the total symmetric ring of quotients $Q^\sigma_{\rm tot}(R)$. We also demonstrate that Morita's construction of $Q^r_{\rm tot}(R)$ can be adapted to the construction of $Q^\sigma_{\rm tot}(R)$. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
18. The Dickson subcategory splitting conjecture for pseudocompact algebras
- Author
-
Iovanov, Miodrag Cristian, Năstăsescu, Constantin, and Torrecillas Jover, Blass
- Subjects
- *
ALGEBRA , *MATHEMATICAL analysis , *MATHEMATICS , *TORSION theory (Algebra) - Abstract
Abstract: Let A be a pseudocompact (or profinite) algebra, so where C is a coalgebra. We show that the if the semiartinian part (the “Dickson” part) of every A-module M splits off in M, then A is semiartinian, giving thus a positive answer in the case of algebras arising as dual of coalgebras (pseudocompact algebras), to a well known conjecture of Faith. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
19. FACTORIZATION, FIBRATION AND TORSION.
- Author
-
Rosický, Jiří and Tholen, Walter
- Subjects
TORSION theory (Algebra) ,HOMOTOPY theory ,MATHEMATICS ,COMMUTATIVE rings ,IDEALS (Algebra) - Abstract
A simple definition of torsion theory is presented, as a factorization system with both classes satisfying the 3-for-2 property. Comparisons with the traditional notion are given, as well as connections with the notions of fibration and of weak factorization system, as used in abstract homotopy theory. [ABSTRACT FROM AUTHOR]
- Published
- 2007
20. G– $$\delta$$ δ –M Modules and Torsion Theory Cogenerated by Such Modules
- Author
-
Behnam Talaee
- Subjects
General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,010103 numerical & computational mathematics ,General Chemistry ,01 natural sciences ,Combinatorics ,Control theory ,Torsion theory ,General Earth and Planetary Sciences ,Homomorphism ,0101 mathematics ,General Agricultural and Biological Sciences ,Mathematics - Abstract
Let N be a module in category $$\sigma [M]$$ . N is called generalized $$\delta$$ –M-small (briefly G– $$\delta$$ –M) if, $$N \subseteq \delta (L)$$ for some $$L \in \sigma [M]$$ . In this paper we characterize G– $$\delta$$ –M modules and get some suitable results related to this kind of modules. We will show that a module $$N \in \sigma [M]$$ is G– $$\delta$$ –M if and only if $$N \subseteq \delta (\hat{N})$$ , where $$\hat{N}$$ is the M-injective envelope of N in $$\sigma [M]$$ . Also we prove that if there is no non-zero G– $$\delta$$ –M module, then M is cosemisimple and by giving an example we show that the converse need not be true. The relation between G– $$\delta$$ –M modules and some other classes of modules would be investigated in this paper. The torsion theory cogenerated by this class of modules will be introduced and studied in this paper. For a module $$N \in \sigma [M]$$ we show that $$N = \mathrm{{Re}}_{GD[M]}(N)$$ iff for every nonzero homomorphism $$f : N \longrightarrow K$$ in $$\sigma [M],$$ $$\mathrm{{Im}}(f) \not \subseteq \delta (K)$$ iff $$\Delta _{\delta }(N,A) = 0,$$ for all $$A \in \sigma [M]$$ .
- Published
- 2017
21. SOME ASPECTS OF MODULAR LATTICES WITH DUAL KRULL DIMENSION.
- Author
-
Teply, Mark L. and Seog Hoon Rim
- Subjects
- *
MODULAR lattices , *LATTICE theory , *TORSION theory (Algebra) , *COMMUTATIVE rings , *IDEALS (Algebra) , *MODULES (Algebra) , *ALGEBRA , *MATHEMATICS - Abstract
For an ordinal α, a modular lattice L with 0 and 1 is α-atomic if L has dual Krull dimension α but each interval [0,x] with x < 1 has dual Krull dimension <α. The properties of α-atomic lattices are presented and applied to module theory. The endomorphism ring of certain types of α-atomic modules is a local domain and hence there is a Krull–Schmidt type theorem for those α-atomic modules. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
22. Comodules and Landweber exact homology theories
- Author
-
Hovey, Mark and Strickland, Neil
- Subjects
- *
ALGEBRA , *MATHEMATICS , *HOMOLOGY theory , *ALGEBRAIC topology - Abstract
We show that, if
E is a commutativeMU -algebra spectrum such thatE* is Landweber exact overMU* , then the category ofE*E -comodules is equivalent to a localization of the category ofMU*MU -comodules. This localization depends only on the heights ofE at the integer primesp . It follows, for example, that the category ofE(n)*E(n) -comodules is equivalent to the category of(vn-1BP)*(vn-1BP) -comodules. These equivalences give simple proofs and generalizations of the Miller–Ravenel and Morava change of rings theorems. We also deduce structural results about the category ofE*E -comodules. We prove that everyE*E -comodule has a primitive, we give a classification of invariant prime ideals inE* , and we give a version of the Landweber filtration theorem. [Copyright &y& Elsevier]- Published
- 2005
- Full Text
- View/download PDF
23. DECOMPOSITIONS OF MODULES SUPPLEMENTED RELATIVE TO A TORSION THEORY.
- Author
-
KOŞAN, M. TAMER and HARMANCI, ABDULLAH
- Subjects
- *
TORSION theory (Algebra) , *ALGEBRA , *MATHEMATICAL combinations , *MATHEMATICAL analysis , *ARITHMETIC , *MATHEMATICS - Abstract
Let R be a ring, M a right R-module and a hereditary torsion theory in Mod-R with associated torsion functor τ for the ring R. Then M is called τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. In [4], M is called almost τ-torsion if every proper submodule of M is τ-torsion. We present here some properties of these classes of modules and look for answers to the following questions posed by the referee of the paper [4]: (1) Let a module M = M′ ⊕ M″ be a direct sum of a semisimple module M′ and τ-supplemented module M″. Is M τ-supplemented? (2) Can one find a non-stable hereditary torsion theory τ and τ-supplemented modules M′ and M″ such that M′ ⊕ M″ is not τ-supplemented? (3) Can one find a stable hereditary torsion theory τ and a τ-supplemented module M such that M/N is not τ-supplemented for some submodule N of M? (4) Let τ be a non-stable hereditary torsion theory and the module M be a finite direct sum of almost τ-torsion submodules. Is M τ-supplemented? (5) Do you know an example of a torsion theory τ and a τ-supplemented module M with τ-torsion submodule τ(M) such that M/τ(M) is not semisimple? [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
24. MODULAR QFD LATTICES WITH APPLICATIONS TO GROTHENDIECK CATEGORIES AND TORSION THEORIES.
- Author
-
Albu, Toma, Iosif, Mihai, and Teply, Mark L.
- Subjects
- *
LATTICE theory , *SET theory , *GROUP theory , *GROTHENDIECK groups , *ALGEBRA , *MATHEMATICS - Abstract
A modular lattice L with 0 and 1 is called quotient finite dimensional (QFD) if [x,1] has no infinite independent set for any x∈L. We extend some results about QFD modules to upper continuous modular lattices by using Lemonnier's Lemma. One result says that QFD for a compactly generated lattice L is equivalent to Condition (C): for every m∈L, there exists a compact element t of L such that t∈[0,m] and [t,m[ has no maximal element. If L is not compactly generated, then QFD and (C) separate into two distinct conditions, which are analyzed and characterized for upper continuous modular lattices. We also extend to upper continuous modular lattices some characterizations of QFD modules with Gabriel dimension. Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory. [ABSTRACT FROM AUTHOR]
- Published
- 2004
- Full Text
- View/download PDF
25. On $\tau$-extending modules
- Author
-
R. Mohammadi and Yahya Talebi
- Subjects
Combinatorics ,Pure mathematics ,Direct sum ,General Mathematics ,Torsion theory ,Filter (mathematics) ,Mathematics::Representation Theory ,Mathematics - Abstract
In this paper we introduce the concept of $\tau$-extending modules by $\tau$-rational submodules and study some properties of such modules. It is shown that the set of all $\tau$-rational left ideals of $_RR$ is a Gabriel filter. An $R$-module $M$ is called $\tau$-extending if every submodule of $M$ is $\tau$-rational in a direct summand of $M$. It is proved that $M$ is $\tau$-extending if and only if $M = Rej_ME(R/\tau(R))\oplus N$, such that $N$ is a $\tau$-extending submodule of $M$. An example is given to show that the direct sum of $\tau$-extending modules need not be $\tau$-extending.
- Published
- 2016
26. Uniqueness of uniform decomposition relative to a torsion theory
- Author
-
Şahin, Eda, Uşak Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, and Şahin, Eda
- Subjects
Matematik ,essentially equivalent ,injective hulls ,Mathematics::Commutative Algebra ,Mathematics::Rings and Algebras ,torsion theory ,uniform module,injective hull,torsion theory ,uniform module ,Krull-Remak-Schmidt Theorem ,Mathematics - Abstract
As a consequence of classical Krull-Remak-Schmidt Theorem, a uniqueness theorem for finite direct sum decomposition into relative uniform modules with local endomorphism rings in torsion theories is reviewed.
- Published
- 2018
27. Pretorsion theories in general categories
- Author
-
Carmelo Antonio Finocchiaro, Marino Gran, Alberto Facchini, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
- Subjects
Pure mathematics ,Commutative Algebra (math.AC) ,Torsion theory ,01 natural sciences ,Morphism ,Non-abelian torsion theory ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Ideal of morphisms ,Category Theory (math.CT) ,0101 mathematics ,Abelian group ,Finite set ,Mathematics ,Subcategory ,Algebra and Number Theory ,010102 general mathematics ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Mathematics - Commutative Algebra ,18E40, 18D20, 17A65, 13D30 ,Category of preordered sets ,Pretorsion theory ,Rings and Algebras (math.RA) ,Torsion (algebra) ,010307 mathematical physics - Abstract
We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair ($\mathcal T$, $\mathcal F$) of full replete subcategories in a category $\mathcal C$, the corresponding full subcategory $\mathcal Z = \mathcal T \cap \mathcal F$ of \emph{trivial objects} in $\mathcal C$. The morphisms which factor through $\mathcal Z$ are called $\mathcal Z$-trivial, and these form an ideal of morphisms, with respect to which one can define $\mathcal Z$-prekernels, $\mathcal Z$-precokernels, and short $\mathcal Z$-preexact sequences. This naturally leads to the notion of pretorsion theory, which is the object of study of this article, and includes the classical one in the abelian context when $\mathcal Z$ is reduced to the $0$-object of $\mathcal C$. We study the basic properties of pretorsion theories, and examine some new examples in the category of all endomappings of finite sets and in the category of preordered sets., 22 pages
- Published
- 2021
28. New results on C 11 and C 12 lattices with applications to Grothendieck categories and torsion theories
- Author
-
Mihai Iosif and Toma Albu
- Subjects
Modular lattice ,Continuation ,Pure mathematics ,Mathematics (miscellaneous) ,Grothendieck category ,Mathematics::Category Theory ,Torsion theory ,010102 general mathematics ,Torsion (algebra) ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions (Ci) 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 (C11) or (C12). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.
- Published
- 2016
29. A Hereditary Torsion Theory for Modules Over Integral Domains and Its Applications
- Author
-
Fanggui Wang and Lei Qiao
- Subjects
Discrete mathematics ,Pure mathematics ,Class (set theory) ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Mathematics::Rings and Algebras ,010102 general mathematics ,Characterization (mathematics) ,01 natural sciences ,Injective function ,Integral domain ,Global dimension ,010101 applied mathematics ,Set (abstract data type) ,Torsion theory ,0101 mathematics ,Flatness (mathematics) ,Mathematics - Abstract
In this article, we study the hereditary torsion theory defined by the set of associated primes of principle ideals of an integral domain, which is called the g-torsion theory. We first discuss some general properties of g-torsion theories, and after that give some applications of them. For example, we generalize a characterization of reflexive modules over quasi-normal domains to a class of non-Noetherian domains. Among other things, a characterization of coherent domains of weak Gorenstein global dimension at most two is also given in terms of Gorenstein projectivity (or Gorenstein flatness) of injective modules relative to the g-torsion theory.
- Published
- 2016
30. A Construction of Totally Reflexive Modules
- Author
-
Janet Striuli, Roger Wiegand, and Hamid Rahmati
- Subjects
Discrete mathematics ,Pure mathematics ,Endomorphism ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Structure (category theory) ,High multiplicity ,01 natural sciences ,Torsion theory ,0103 physical sciences ,Bimodule ,010307 mathematical physics ,0101 mathematics ,Indecomposable module ,Mathematics - 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.
- Published
- 2016
31. Lattice preradicals with applications to Grothendieck categories and torsion theories
- Author
-
Mihai Iosif and Toma Albu
- Subjects
Algebra ,Modular lattice ,Algebra and Number Theory ,Complete lattice ,Mathematics::K-Theory and Homology ,Grothendieck category ,Mathematics::Category Theory ,Torsion theory ,Torsion (algebra) ,Jacobson radical ,Mathematics - 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.
- Published
- 2015
32. A framework for torsion theory computations on elliptic threefolds
- Author
-
Jason Lo, David Angeles, and Courtney M. Van Der Linden
- Subjects
Pure mathematics ,14F05 (Primary) 18E40, 14J30 (Secondary) ,General Mathematics ,Computation ,010102 general mathematics ,Mathematics - Category Theory ,01 natural sciences ,Coherent sheaf ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Mathematics::K-Theory and Homology ,Torsion theory ,0103 physical sciences ,Torsion (algebra) ,FOS: Mathematics ,Homological algebra ,Category Theory (math.CT) ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Topology (chemistry) ,Mathematics - 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., Comment: 15 pages
- Published
- 2018
- Full Text
- View/download PDF
33. Using torsion theory to compute the algebraic structure of Hochschild (co)homology
- Author
-
Estanislao Herscovich, Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Institut Fourier (IF), and Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)
- Subjects
Pure mathematics ,Algebraic structure ,010102 general mathematics ,Homology (mathematics) ,16. Peace & justice ,01 natural sciences ,Mathematics (miscellaneous) ,Torsion theory ,0103 physical sciences ,[MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT] ,Homological algebra ,010307 mathematical physics ,Koszul algebra ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
International audience
- Published
- 2018
34. Minimal inclusions of torsion classes
- Author
-
Shijie Zhu, Emily Barnard, and Andrew T. Carroll
- Subjects
Pure mathematics ,Representation theory ,Mathematics::K-Theory and Homology ,05E10, 06B15 ,Torsion theory ,Associative algebra ,Torsion (algebra) ,Bijection ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Representation Theory (math.RT) ,Indecomposable module ,Quotient ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $\Lambda$ be a finite-dimensional associative algebra. The torsion classes of $mod\, \Lambda$ form a lattice under containment, denoted by $tors\, \Lambda$. In this paper, we characterize the cover relations in $tors\, \Lambda$ by certain indecomposable modules. We consider three applications: First, we show that the completely join-irreducible torsion classes (torsion classes which cover precisely one element) are in bijection with bricks. Second, we characterize faces of the canonical join complex of $tors\, \Lambda$ in terms of representation theory. Finally, we show that, in general, the algebra $\Lambda$ is not characterized by its lattice $tors\, \Lambda$. In particular, we study the torsion theory of a quotient of the preprojective algebra of type $A_n$. We show that its torsion class lattice is isomorphic to the weak order on $A_n$., Comment: 25 pages, 10 figures
- Published
- 2017
35. MTame Modules And Local Gabriel Correspondence
- Author
-
Jaime Castro Pérez and José Ríos Montes
- Subjects
Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Dimension (vector space) ,Torsion theory ,Krull dimension ,Prime (order theory) ,Mathematics - Abstract
Using the concept of prime submodule for M ∈ R-Mod, P ∈ Spec(M), and N ∈ σ[M], we define when N is P-Mtame \ (Mtame) module. This concept generalizes the concept\ of P-tame (tame) modules. For M ∈ R-Mod and τ ∈M-tors, we use the concept of τ M -Gabriel dimension and we study the relationship between Mtame modules and τ M -Gabriel dimension. We find equivalent conditions for a module M progenerator in σ[M] with τ M -Gabriel dimension to have τ M -Gabriel correspondence in terms of the P-Mtame modules. This result extends the results by Albu et al. and Kim and Krause.
- Published
- 2015
36. Non-periodic homogenization of bending–torsion theory for inextensible rods from 3D elasticity
- Author
-
Igor Velčić and Maroje Marohnić
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Limiting ,01 natural sciences ,Homogenization (chemistry) ,Rod ,010101 applied mathematics ,Dimensional reduction ,Torsion theory ,Subsequence ,0101 mathematics ,Nonlinear elasticity ,Mathematics - Abstract
We derive, by means of $$\Gamma $$ -convergence, the equations of homogenized bending rod starting from 3D nonlinear elasticity equations. The main assumption is that the energy behaves like $$h^2$$ (after dividing by $$h^2$$ , the order of vanishing volume), where h is the thickness of the body. We do not presuppose any kind of periodicity and work in the general framework. The result shows that, on a subsequence, we always obtain the equations of the same type as in bending–torsion rod theory and identifies, in an abstract formulation, the limiting quadratic form connected with that model. This result is the generalization of periodic homogenization of bending–torsion rod theory already present in the literature.
- Published
- 2015
37. Torsion Theory and its Applications in M − D Modules
- Author
-
Behnam Talaee
- Subjects
Statistics and Probability ,Economics and Econometrics ,Ring (mathematics) ,Pure mathematics ,Mathematics::Commutative Algebra ,Generalization ,Torsion theory ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Let R be a ring and M an R−module. A module N ∈ (M ) is called M-small if, N ≪ K for some K ∈ (M ). Torsion theory cogenerated by M −small modules is introduced and investigated in (9). Also as a generalization of M −small modules, − M −small modules are studied in (6). In this paper we will introduce M −delta (briefly M − D) modules and investigate the torsion theory cogenerated by such modules . We will get some equivalent conditions for when N is equal to its torsion theory submodule cogenerated by M − D modules. Especially we show that D(N;A) = 0 for all A ∈ (M ) iff N = ReD(M)(N ). Some other important properties about this kind of modules will be obtained.
- Published
- 2014
38. Locally torsion-free quasi-coherent sheaves
- Author
-
Sinem Odabasi
- Subjects
Pure mathematics ,13D30, 18E40, 18F20, 14F05, 18A30 ,Algebra and Number Theory ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Coherent sheaf ,Mathematics - Algebraic Geometry ,Rings and Algebras (math.RA) ,Mathematics::Category Theory ,Torsion theory ,FOS: Mathematics ,Torsion (algebra) ,Category Theory (math.CT) ,Dedekind cut ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Let $X$ be an arbitrary scheme. The category $\mathfrak{Qcoh}(X)$ of quasi--coherent sheaves on $X$ is known that admits arbitrary direct products. However their structure seems to be rather mysterious. In the present paper we will describe the structure of the product object of a family of locally torsion-free objects in $\mathfrak{Qcoh}(X)$, for $X$ an integral scheme. Several applications are provided. For instance it is shown that the class of flat quasi--coherent sheaves on a Dedekind scheme $X$ is closed under arbitrary direct products, and that the class of all locally torsion-free quasi--coherent sheaves induces a hereditary torsion theory on $\mathfrak{Qcoh}(X)$. Finally torsion-free covers are shown to exist in $\mathfrak{Qcoh}(X)$.
- Published
- 2014
39. A Factorization Theorem for Topological Abelian Groups
- Author
-
Anna Giordano Bruno and Dikran Dikranjan
- Subjects
Algebra and Number Theory ,Torsion subgroup ,G-module ,abelian group ,bounded abelian group ,compact abelian group ,divisible weight ,factorization theorem ,torsion theory ,w-divisible group ,Elementary abelian group ,Topology ,Rank of an abelian group ,Free abelian group ,Non-abelian group ,Abelian group ,Arithmetic of abelian varieties ,Mathematics - Abstract
Using the nice properties of the w-divisible weight and the w-divisible groups, we prove a factorization theorem for compact abelian groups K; namely, K = K tor × K d , where K tor is a bounded torsion compact abelian group and K d is a w-divisible compact abelian group. By Pontryagin duality this result is equivalent to the same factorization for discrete abelian groups proved in [9].
- Published
- 2014
40. Two applications of Nagata rings and modules
- Author
-
Lei Qiao and Fanggui Wang
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Applied Mathematics ,Torsion theory ,010102 general mathematics ,010103 numerical & computational mathematics ,Commutative ring ,0101 mathematics ,Type (model theory) ,01 natural sciences ,Flat module ,Mathematics - Abstract
Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].
- Published
- 2019
41. Monotone-light factorisation systems and torsion theories
- Author
-
Marino Gran, Tomas Everaert, and Analytical, Categorical and Algebraic Topology
- Subjects
Subcategory ,Pure mathematics ,Factorisation system ,General Mathematics ,Totally disconnected group ,Mathematics - Category Theory ,Reduced ring ,Commutative ring ,Normal category ,Torsion theory ,Torsion-free group ,18E40, 18A40, 16S90, 16N80, 20K15, 54H11 ,Monotone polygon ,Mathematics::Category Theory ,FOS: Mathematics ,Torsion (algebra) ,Category Theory (math.CT) ,Abelian category ,Topological group ,Abelian group ,Monotone-light factorisation ,Quotient ,Mathematics - Abstract
Given a torsion theory (Y,X) in an abelian category C, the reflector I from C to the torsion-free subcategory X induces a reflective factorisation system (E, M) on C. It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Par\'e that (E, M) induces a monotone-light factorisation system (E',M*) by simultaneously stabilising E and localising M, whenever the torsion theory is hereditary and any object in C is a quotient of an object in X. 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., Comment: 12 pages
- Published
- 2013
42. ℱ-Noetherian and Weakly Exact Torsion Theories
- Author
-
Ladislav Bican
- Subjects
Discrete mathematics ,Noetherian ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Mathematics::K-Theory and Homology ,Torsion theory ,Mathematics::Rings and Algebras ,Torsion (algebra) ,Injective function ,Mathematics - Abstract
It is well-known (see [13]) that a hereditary torsion theory τ for the category R-mod is noetherian if and only if the class of all τ-torsionfree τ-injective modules is closed under arbitrary direct sums. So, it is natural to investigate the hereditary torsion theories having the property that the class of all τ-torsionfree injective modules is closed under arbitrary direct sums, which are called ℱ-noetherian. These torsion theories have been studied by Teply in [16]. In the second part of this note we shall study the weakly exact hereditary torsion theories, which generalize the exact one's.
- Published
- 2013
43. NEAT AND CONEAT SUBMODULES OF MODULES OVER COMMUTATIVE RINGS
- Author
-
Septimiu Crivei
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Torsion theory ,Closure (topology) ,Maximal ideal ,Commutative ring ,Injective function ,Mathematics - Abstract
We prove that neat and coneat submodules of a module coincide when $R$ is a commutative ring such that every maximal ideal is principal, extending a recent result by Fuchs. We characterise absolutely neat (coneat) modules and study their closure properties. We show that a module is absolutely neat if and only if it is injective with respect to the Dickson torsion theory.
- Published
- 2013
44. Perverse Coherent t-Structures Through Torsion Theories
- Author
-
Jorge Vitória
- Subjects
Perverse coherent sheaves ,0209 industrial biotechnology ,Pure mathematics ,General Mathematics ,Noncommutative projective planes ,t-structure ,Torsion theory ,02 engineering and technology ,01 natural sciences ,Coherent sheaf ,Mathematics::Algebraic Geometry ,020901 industrial engineering & automation ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,FOS: Mathematics ,Category Theory (math.CT) ,Representation Theory (math.RT) ,0101 mathematics ,Projective test ,Mathematics::Representation Theory ,Mathematics ,Derived category ,010102 general mathematics ,Mathematics - Category Theory ,16. Peace & justice ,Noncommutative geometry ,Torsion (algebra) ,Projective plane ,Mathematics - Representation Theory ,14F05, 13D09, 13D30, 16E35, 16S38, 16W50 - Abstract
Bezrukavnikov (later together with Arinkin) recovered the work of Deligne defining perverse $t$-structures for the derived category of coherent sheaves on a projective variety. In this text we prove that these $t$-structures can be obtained through tilting torsion theories as in the work of Happel, Reiten and Smal��. This approach proves to be slightly more general as it allows us to define, in the quasi-coherent setting, similar perverse $t$-structures for certain noncommutative projective planes., New revised version with important corrections
- Published
- 2013
45. Semiartinian Profinite Algebras have Nilpotent Jacobson Radical
- Author
-
Miodrag Cristian Iovanov
- Subjects
Mathematics::Group Theory ,Nilpotent ,Pure mathematics ,General Mathematics ,Coalgebra ,Torsion theory ,Mathematics::Rings and Algebras ,Filtration (mathematics) ,Jacobson radical ,Algebra over a field ,Mathematics - 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, 2008).
- Published
- 2013
46. M-purity and torsion purity in modules
- Author
-
Ashok Kumar Pandey, Ewing Christian, and Manoj Pathak
- Subjects
Pure mathematics ,Torsion theory ,Torsion (algebra) ,Mathematics - Abstract
The aim of this paper is to relativize the concept of M- purity and σ- purity defined and studied by Azumaya[2] with respect to an arbitrary hereditary torsion theory given by a left exact torsion redical σ and also relate this concepts with the notions of σ- purity as given by B. B. Bhattacharya and D. P. Choudhury[3] and Ashok Kr. Pandey[1].
- Published
- 2013
47. ON COMPLETELY PRIME SUBMODULES
- Author
-
David Ssevviiri
- Subjects
Matematik ,Algebra and Number Theory ,Mathematics - Rings and Algebras ,Prime (order theory) ,Domain,prime module,completely prime module,completely prime radical,torsion theory ,Combinatorics ,Section (category theory) ,Rings and Algebras (math.RA) ,Torsion theory ,Domain (ring theory) ,16S90, 16D60, 16D99 ,Torsion (algebra) ,FOS: Mathematics ,Algebra over a field ,Mathematics - Abstract
The formal study of completely prime modules was initiated by N. J. Groenewald and the current author in the paper; Completely prime submodules, {\it Int. Elect. J. Algebra}, {\bf 13}, (2013), 1--14. In this paper, the study of completely prime modules is continued. Firstly, the advantage completely prime modules have over prime modules is highlited and different situations that lead to completely prime modules given. Later, emphasis is put on fully completely prime modules, (i.e., modules whose all submodules are completely prime). For a fully completely prime left $R$-module $M$, if $a, b\in R$ and $m\in M$, then $abm=bam$, $am=a^km$ for all positive integers $k$, and either $am=abm$ or $bm=abm$. In the last section, two different torsion theories induced by the completely prime radical are given., Comment: 15 pages
- Published
- 2016
48. Torsion classes generated by silting modules
- Author
-
Jan Žemlička and Simion Breaz
- Subjects
18G15 ,cosilting ,General Mathematics ,010102 general mathematics ,preenveloping class ,silting ,torsion theory ,010103 numerical & computational mathematics ,Mathematics - Rings and Algebras ,01 natural sciences ,16E30 ,Combinatorics ,Rings and Algebras (math.RA) ,Torsion theory ,Mathematics::Category Theory ,16D90 ,FOS: Mathematics ,Torsion (algebra) ,Representation Theory (math.RT) ,0101 mathematics ,precovering class ,Mathematics - Representation Theory ,Mathematics - 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 $\mathcal{T}$ such that the regular module has a special $\mathcal{T}$-preenvelope. In particular every torsion enveloping class in $\textrm{Mod-} R$ are of the form $\mathrm{Gen}(T)$ 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 $\mathrm{Cogen}(T)$, where $T$ is a cosilting module., Preliminary version; comments are welcome; v2: improved version; an important gap in the initial version was completed; v3: We added some examples; references added and updated
- Published
- 2016
49. Rigorous Derivation of a Homogenized Bending-Torsion Theory for Inextensible Rods from Three-Dimensional Elasticity
- Author
-
Stefan Neukamm
- Subjects
Mathematics (miscellaneous) ,Mechanical Engineering ,Torsion theory ,Mathematical analysis ,Bending ,Elasticity (economics) ,Analysis ,Rod ,Mathematics - Published
- 2012
50. Prime Submodules and Local Gabriel Correspondence in σ[M]
- Author
-
Jaime Castro Pérez and José Ríos Montes
- Subjects
Associated prime ,Combinatorics ,Pure mathematics ,Algebra and Number Theory ,Dimension (vector space) ,Torsion theory ,Isomorphism ,Indecomposable module ,Injective function ,Prime (order theory) ,Mathematics - 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.
- Published
- 2012
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.