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Journal articles on the topic 'Torsion theory (algebra)'

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Published: 4 June 2021

Last updated: 5 March 2023

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1

PRICE, KENNETH L. "A DOMAIN TEST FOR LIE COLOR ALGEBRAS." Journal of Algebra and Its Applications 07, no. 01 (February 2008): 81–90. http://dx.doi.org/10.1142/s0219498808002679.

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Lie color algebras are generalizations of Lie superalgebras and graded Lie algebras. The properties of a Lie color algebra can often be related directly to the ring structure of its universal enveloping algebra. We study the effects of torsion elements and torsion subspaces. Let [Formula: see text] denote a Lie color algebra. If [Formula: see text] is homogeneous and torsion then x2 = 0 in [Formula: see text]. If no homogeneous element of [Formula: see text] is torsion, then [Formula: see text] so [Formula: see text] is semiprime. In this case we can give a test which uses Gröbner basis methods to determine when [Formula: see text] is a domain. This is applied in an example to show [Formula: see text] may be a domain even if [Formula: see text] contains torsion elements and torsion subspaces.

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2

Okoh, F. "Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras." Canadian Mathematical Bulletin 39, no. 1 (March 1, 1996): 111–14. http://dx.doi.org/10.4153/cmb-1996-014-9.

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AbstractIf R is a Dedekind domain, then div splits i.e.; the maximal divisible submodule of every R-module M is a direct summand of M. We investigate the status of this result for some finite-dimensional hereditary algebras. We use a torsion theory which permits the existence of torsion-free divisible modules for such algebras. Using this torsion theory we prove that the algebras obtained from extended Coxeter- Dynkin diagrams are the only such hereditary algebras for which div splits. The field of rational functions plays an essential role. The paper concludes with a new type of infinite-dimensional indecomposable module over a finite-dimensional wild hereditary algebra.

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3

BARLAK, SELÇUK, TRON OMLAND, and NICOLAI STAMMEIER. "On the -theory of -algebras arising from integral dynamics." Ergodic Theory and Dynamical Systems 38, no. 3 (September 22, 2016): 832–62. http://dx.doi.org/10.1017/etds.2016.63.

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We investigate the$K$-theory of unital UCT Kirchberg algebras${\mathcal{Q}}_{S}$arising from families$S$of relatively prime numbers. It is shown that$K_{\ast }({\mathcal{Q}}_{S})$is the direct sum of a free abelian group and a torsion group, each of which is realized by another distinct$C^{\ast }$-algebra naturally associated to$S$. The$C^{\ast }$-algebra representing the torsion part is identified with a natural subalgebra${\mathcal{A}}_{S}$of${\mathcal{Q}}_{S}$. For the$K$-theory of${\mathcal{Q}}_{S}$, the cardinality of$S$determines the free part and is also relevant for the torsion part, for which the greatest common divisor$g_{S}$of$\{p-1:p\in S\}$plays a central role as well. In the case where$|S|\leq 2$or$g_{S}=1$we obtain a complete classification for${\mathcal{Q}}_{S}$. Our results support the conjecture that${\mathcal{A}}_{S}$coincides with$\otimes _{p\in S}{\mathcal{O}}_{p}$. This would lead to a complete classification of${\mathcal{Q}}_{S}$, and is related to a conjecture about$k$-graphs.

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4

Bland, Paul E. "Differential torsion theory." Journal of Pure and Applied Algebra 204, no. 1 (January 2006): 1–8. http://dx.doi.org/10.1016/j.jpaa.2005.03.005.

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5

Robertson, Guyan, and Tim Steger. "AsymptoticK-Theory for Groups Acting onÃ2Buildings." Canadian Journal of Mathematics 53, no. 4 (August 1, 2001): 809–33. http://dx.doi.org/10.4153/cjm-2001-033-4.

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AbstractLet Γ be a torsion free lattice inG= PGL(3,) whereis a nonarchimedean local field. Then Γ acts freely on the affine Bruhat-Tits building ℬ ofGand there is an induced action on the boundary Ω of ℬ. The crossed productC*-algebra(Γ) =C(Ω) ⋊ Γ depends only on Γ and is classified by itsK-theory. This article shows how to compute theK-theory of(Γ) and of the larger class of rank two Cuntz-Krieger algebras.

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6

Kaufmann, Ralph M. "The algebra of discrete torsion." Journal of Algebra 282, no. 1 (December 2004): 232–59. http://dx.doi.org/10.1016/j.jalgebra.2004.07.042.

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7

Adachi, Takahide, Osamu Iyama, and Idun Reiten. "-tilting theory." Compositio Mathematica 150, no. 3 (December 3, 2013): 415–52. http://dx.doi.org/10.1112/s0010437x13007422.

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AbstractThe aim of this paper is to introduce $\tau $-tilting theory, which ‘completes’ (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field $k$ is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras $kQ$, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) $\tau $-tilting modules, and show that an almost complete support $\tau $-tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional $k$-algebra $\Lambda $, we establish bijections between functorially finite torsion classes in $ \mathsf{mod} \hspace{0.167em} \Lambda $, support $\tau $-tilting modules and two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$. Moreover, these objects correspond bijectively to cluster-tilting objects in $ \mathcal{C} $ if $\Lambda $ is a 2-CY tilted algebra associated with a 2-CY triangulated category $ \mathcal{C} $. As an application, we show that the property of having two complements holds also for two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$.

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8

TRUINI, P., and V. S. VARADARAJAN. "QUANTIZATION OF REDUCTIVE LIE ALGEBRAS: CONSTRUCTION AND UNIVERSALITY." Reviews in Mathematical Physics 05, no. 02 (June 1993): 363–415. http://dx.doi.org/10.1142/s0129055x93000103.

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The present paper addresses the question of universality of the quantization of reductive Lie algebras. Quantization is viewed as a torsion free deformation depending upon several parameters which are treated formally and not as complex numbers. The coalgebra and algebra structures are shown to restrict very sharply the possibilities for the infinite series in the generators of the Cartan subalgebra. Under an Ansatz which can be viewed as requiring that the two Borel subalgebras are deformed as Hopf algebras we construct a multi-parameter quantization which has the required property of universality. We also show that such a quantization can be defined so that the algebra structure is the same as that of the standard one-parameter quantization, the remaining parameters being relegated to the coalgebra structure. We discuss an example in which only the latter parameters appear in the deformation. We then complete the study of the universal deformations by developing some aspects of the representation theory of the deformed algebras. Using this theory, especially the freeness of the irreducible modules, we prove the analogue of the Poincaré-Birkhoff-Witt theorem, and, as a consequence, the torsion freeness of the universal deformations.

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9

Li, Hou Guo. "Application of Lie Group Analysis to the Plastic Torsion of Rod with the Saint Venant–Mises Yield Criterion." Advanced Materials Research 461 (February 2012): 265–71. http://dx.doi.org/10.4028/www.scientific.net/amr.461.265.

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Based on Lie group and Lie algebra theory, the basic principles of Lie group analysis of differential equations in mechanics are analyzed, and its validity in theory of plasticity is explained by example. For the plastic torsion of rod with variable cross section that consists in non-linear Saint Venant-Mises yield criterion, the 10-dimensional Lie algebra admitted by the equilibrium equation and yield criterion is completely solved, and invariants and group invariant solutions relative to different sub-algebras are given. At last, physical explanations of each group invariant solution are discussed by some types of transformations.

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10

Colby, R. R., and K. R. Fuller. "Hereditary Torsion Theory Counter Equivalences." Journal of Algebra 183, no. 1 (July 1996): 217–30. http://dx.doi.org/10.1006/jabr.1996.0215.

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11

Magalhães, L. "Some Results in the Connective K-Theory of Lie Groups." Canadian Mathematical Bulletin 31, no. 2 (June 1, 1988): 194–99. http://dx.doi.org/10.4153/cmb-1988-030-9.

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AbstractIn this paper we give a description of:(1) the Hopf algebra structure of k*(G; L) when G is a compact, connected Lie group and L is a ring of type Q(P) so that H*(G; L) is torsion free;(2) the algebra structure of k*(G2; L) for L = Z2 or Z.

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12

Nasrollah Nejad, Abbas, Zahra Shahidi, and Rashid Zaare-Nahandi. "Torsion-free Aluffi algebras." Journal of Algebra 513 (November 2018): 190–207. http://dx.doi.org/10.1016/j.jalgebra.2018.07.024.

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13

MIGNEMI, S. "LINEAL GRAVITY WITH DYNAMICAL TORSION." Modern Physics Letters A 11, no. 15 (May 20, 1996): 1235–45. http://dx.doi.org/10.1142/s0217732396001259.

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We investigate the string-inspired action for two-dimensional gravity with the addition of dynamical torsion and obtain the most general static black hole solutions. We also consider the Hamiltonian formulation of the model and discuss its symmetries, showing that it can be considered as a gauge theory of a nonlinear generalization of the two-dimensional Poincare algebra. Finally, we briefly discuss the quantization of the theory in the Dirac formalism.

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14

Lampret, Leon, and Aleš Vavpetič. "Torsion table for the lie algebra niln." Communications in Algebra 47, no. 9 (April 4, 2019): 3567–78. http://dx.doi.org/10.1080/00927872.2019.1567751.

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15

Chiodo, Maurice, and Rishi Vyas. "Torsion, torsion length and finitely presented groups." Journal of Group Theory 21, no. 5 (September 1, 2018): 949–71. http://dx.doi.org/10.1515/jgth-2018-0022.

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Abstract We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some {C^{\prime}(1/6)} finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is {C^{\prime}(1/6)} , and thus word-hyperbolic and virtually torsion-free.

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16

Colby, R. R., and K. R. Fuller. "Tilting and torsion theory counter equivalences." Communications in Algebra 23, no. 13 (January 1995): 4833–49. http://dx.doi.org/10.1080/00927879508825503.

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17

Wiegandt, Richard. "Radical and Torsion Theory for Acts." Semigroup Forum 72, no. 2 (March 15, 2006): 312–28. http://dx.doi.org/10.1007/s00233-005-0546-5.

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18

Eckhardt, Caleb, and Paul McKenney. "Finitely generated nilpotent group C*-algebras have finite nuclear dimension." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 738 (May 1, 2018): 281–98. http://dx.doi.org/10.1515/crelle-2015-0049.

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Abstract We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra A generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, A satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its ordered K-theory and is approximately subhomogeneous. We observe that all C*-algebras generated by faithful irreducible representations of finitely generated, torsion free nilpotent groups satisfy the universal coefficient theorem.

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19

van den Berg, John E. "When every torsion preradical is a torsion radical." Communications in Algebra 27, no. 11 (January 1999): 5527–47. http://dx.doi.org/10.1080/00927879908826771.

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20

Spiegel, Eugene. "Torsion Elements in Incidence Algebras." Communications in Algebra 31, no. 10 (January 10, 2003): 5179–89. http://dx.doi.org/10.1081/agb-120023153.

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21

Spiegel, E. "Torsion Subgroups of Incidence Algebras." Communications in Algebra 34, no. 5 (June 2006): 1891–96. http://dx.doi.org/10.1080/00927870500542846.

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22

CHARALAMBIDES, STELIOS, and JOHN CLARK. "MAX MODULES RELATIVE TO A TORSION THEORY." Journal of Algebra and Its Applications 07, no. 01 (February 2008): 21–45. http://dx.doi.org/10.1142/s021949880800259x.

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We introduce the concepts of τ-Max modules and left τ-Max rings which are torsion-theoretic analogues of Max modules and left Max rings. A generalization is obtained of an important theorem by Shock and used to characterize τ-Noetherian rings using τ-Max modules. We then characterize left τ-Max rings and obtain a torsion-theoretic version of a result by Hirano. We conclude with results on τ-short modules, introduced as a torsion analogue of a concept recently defined by Bilhan and Smith.

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23

Baker, Matthew H., and Kenneth A. Ribet. "Galois theory and torsion points on curves." Journal de Théorie des Nombres de Bordeaux 15, no. 1 (2003): 11–32. http://dx.doi.org/10.5802/jtnb.384.

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24

Xin, Lix. "Small submodules relative to a torsion theory." Communications in Algebra 21, no. 1 (January 1993): 13–23. http://dx.doi.org/10.1080/00927879208824547.

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25

Picado, Jorge. "On two extension of dicksons torsion theory." Communications in Algebra 21, no. 8 (January 1993): 2749–69. http://dx.doi.org/10.1080/00927879308824704.

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26

Zhu, Xiaosheng. "Torsion theory extensions and finite normalizing extensions." Journal of Pure and Applied Algebra 176, no. 2-3 (December 2002): 259–73. http://dx.doi.org/10.1016/s0022-4049(02)00116-0.

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27

Bueso, José L., and Pascual Jara. "Semiartinian modules relative to a torsion theory." Communications in Algebra 13, no. 3 (January 1985): 631–44. http://dx.doi.org/10.1080/00927878508823181.

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28

García Hernández, J. L., J. L. Gómez Pardo, and J. Martínez Hernández. "Semiperfect modules relative to a torsion theory." Journal of Pure and Applied Algebra 43, no. 2 (December 1986): 145–72. http://dx.doi.org/10.1016/0022-4049(86)90092-7.

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29

McLENDON, MICHAEL. "DETECTING TORSION IN SKEIN MODULES USING HOCHSCHILD HOMOLOGY." Journal of Knot Theory and Its Ramifications 15, no. 02 (February 2006): 259–77. http://dx.doi.org/10.1142/s0218216506004440.

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Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.

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30

Derickx, Maarten, Anastassia Etropolski, Mark van Hoeij, Jackson S. Morrow, and David Zureick-Brown. "Sporadic cubic torsion." Algebra & Number Theory 15, no. 7 (November 1, 2021): 1837–64. http://dx.doi.org/10.2140/ant.2021.15.1837.

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31

Albrecht, Ulrich, and Stefan Friedenberg. "Torsion-Free Ext." Communications in Algebra 44, no. 9 (May 19, 2016): 3976–88. http://dx.doi.org/10.1080/00927872.2015.1095029.

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32

Clark, Pete L., Marko Milosevic, and Paul Pollack. "Typically bounding torsion." Journal of Number Theory 192 (November 2018): 150–67. http://dx.doi.org/10.1016/j.jnt.2018.04.005.

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33

Brandal, Willy, and Erol Barbut. "Decomposing torsion modules." Communications in Algebra 15, no. 6 (January 1987): 1109–17. http://dx.doi.org/10.1080/00927878708823459.

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34

Assem, Ibrahim, and Otto Kerner. "Constructing Torsion Pairs." Journal of Algebra 185, no. 1 (October 1996): 19–41. http://dx.doi.org/10.1006/jabr.1996.0310.

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35

CHUNG, JAE-WOOK, and XIAO-SONG LIN. "TORSION OF QUASI-ISOMORPHISMS." Journal of Knot Theory and Its Ramifications 18, no. 09 (September 2009): 1227–58. http://dx.doi.org/10.1142/s0218216509007440.

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In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a based chain complex to another. It shares nice properties with torsion of acyclic based chain complexes, like multiplicativity and duality. We will further generalize our torsion to quasi-isomorphisms between free chain complexes over a ring under some mild condition. We anticipate that the study of torsion of quasi-isomorphisms will be fruitful in many directions, and in particular, in the study of links in 3-manifolds.

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36

Klingler, Lee, and Lawrence S. Levy. "Wild torsion modules over Weyl algebras and general torsion modules over HNPs." Journal of Algebra 172, no. 2 (March 1995): 273–300. http://dx.doi.org/10.1016/s0021-8693(05)80003-1.

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37

Sánchez–Mirafuentes, Marco Antonio, Julio Cesar Salas–Torres, and Gabriel Villa–Salvador. "Cogalois theory and drinfeld modules." Journal of Algebra and Its Applications 19, no. 01 (January 10, 2019): 2050001. http://dx.doi.org/10.1142/s0219498820500012.

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In this paper, we generalize the results of [M. Sánchez-Mirafuentes and G. Villa–Salvador, Radical extensions for the Carlitz module, J. Algebra 398 (2014) 284–302] to rank one Drinfeld modules with class number one. We show that, in the present form, there does not exist a cogalois theory for Drinfeld modules of rank or class number larger than one. We also consider the torsion of the Carlitz module for the extension [Formula: see text].

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38

DAUNS, JOHN. "NATURAL CLASSES AND TORSION THEORIES." Journal of Algebra and Its Applications 02, no. 01 (March 2003): 85–99. http://dx.doi.org/10.1142/s0219498803000441.

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Conditions under which a natural class of right R-modules is closed under quotient modules are determined. In this case the unique complementary natural class of c(Δ) is closed under direct products. Hence (Δ, c(Δ)) is a hereditary torsion theory with torsion class Δ. In general a natural class [Formula: see text] of right R-modules is not closed under direct products (or quotient modules), yet it has been shown by Y. Zhou that each natural class [Formula: see text] determines a unique hereditary torsion theory τ. The torsion and torsion free classes of this torsion theory τ are studied, and in particular, their dependence on the original natural class [Formula: see text]. As an application, the resulting torsion theories τ are used to define the class of τ-simple, i.e., τ-cocritical modules. Most of the above is done more generally for M-natural classes in the category σ[M].

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39

Stöhr, Ralph. "On torsion in free central extensions of some torsion-free groups." Journal of Pure and Applied Algebra 46, no. 2-3 (June 1987): 249–89. http://dx.doi.org/10.1016/0022-4049(87)90096-x.

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40

Abdollahi, A., and S. M. Zanjanian. "Nonunits of group algebras over the fours group." Journal of Algebra and Its Applications 18, no. 12 (November 3, 2019): 1950228. http://dx.doi.org/10.1142/s0219498819502281.

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The conjecture on units of group algebras of a torsion-free supersoluble group is saying that every unit is trivial, i.e. a product of a nonzero element of the field and an element of the group. This conjecture is still open and even in the slightly simple case of the fours group [Formula: see text], it is not yet known. The main result of this paper is to show that a wide range of elements of group algebra of [Formula: see text] are nonunit.

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41

Bican, Ladislav, Robert El basher, and Blas Torrecillas. "Kulikov's criterion for a non-hereditary torsion theory." Communications in Algebra 23, no. 4 (January 1995): 1275–82. http://dx.doi.org/10.1080/00927879508825280.

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42

Smith, Patrick F., Ana M. De Viola- Prioli, and Jorge E. Viola- Prioli. "Modules complemented with respect to a torsion theory." Communications in Algebra 25, no. 4 (January 1997): 1307–26. http://dx.doi.org/10.1080/00927879708825921.

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43

Çeken, Seçil. "Comultiplication modules relative to a hereditary torsion theory." Communications in Algebra 47, no. 10 (March 23, 2019): 4283–96. http://dx.doi.org/10.1080/00927872.2019.1586910.

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44

Richman, Fred. "The constructive theory of torsion-free abelian groups." Communications in Algebra 18, no. 11 (January 1990): 3913–22. http://dx.doi.org/10.1080/00927879008824116.

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45

Colpi, Riccardo, Enrico Gregorio, and Francesca Mantese. "On the Heart of a faithful torsion theory." Journal of Algebra 307, no. 2 (January 2007): 841–63. http://dx.doi.org/10.1016/j.jalgebra.2006.01.020.

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46

Dudakov, Sergey Mikhailovich. "On Undecidability of Finite Subsets Theory for Torsion Abelian Groups." Mathematics 10, no. 3 (February 8, 2022): 533. http://dx.doi.org/10.3390/math10030533.

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Let M be a commutative cancellative monoid with an element of infinite order. The binary operation can be extended to all finite subsets of M by the pointwise definition. So, we can consider the theory of finite subsets of M. Earlier, we have proved the following result: in the theory of finite subsets of M elementary arithmetic can be interpreted. In particular, this theory is undecidable. For example, the free monoid (the sets of all words with concatenation) has this property, the corresponding algebra of finite subsets is the theory of all finite languages with concatenation. Another example is an arbitrary Abelian group that is not a torsion group. But the method of proof significantly used an element of infinite order, hence, it can’t be immediately generalized to torsion groups. In this paper we prove the given theorem for Abelian torsion groups that have elements of unbounded order: for such group, the theory of finite subsets allows interpreting the elementary arithmetic.

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47

Helme-Guizon, Laure, Józef H. Przytycki, and Yongwu Rong. "Torsion in graph homology." Fundamenta Mathematicae 190 (2006): 139–77. http://dx.doi.org/10.4064/fm190-0-5.

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48

Shumakovitch, Alexander N. "Torsion of Khovanov homology." Fundamenta Mathematicae 225, no. 1 (2014): 343–64. http://dx.doi.org/10.4064/fm225-1-16.

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49

Akhtar, Reza. "Torsion in MixedK-Groups." Communications in Algebra 32, no. 1 (March 2004): 295–313. http://dx.doi.org/10.1081/agb-120027868.

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50

Bhatt, Bhargav. "Torsion completions are bounded." Journal of Pure and Applied Algebra 223, no. 5 (May 2019): 1940–45. http://dx.doi.org/10.1016/j.jpaa.2018.08.008.

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