José Gómez Pardo, Pedro A. Guil Asensio · Proceedings of the American Mathematical Society · 2000
We show that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"> <mml:semantics> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:annotation encoding="application/x-tex">\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -CS module is a direct sum of uniform modules, thus solving an open problem posed in 1994 by Dung, Huynh, Smith and Wisbauer. With the help of this result we also answer several other questions related to indecomposable decompositions of CS-modules.
Cited by 21Open access
Hamel, Josée · Knowledge UdeS (Institutional Deposit of the University of Sherbrooke) · 1996
Soient K un corps algébriquement clos et A une K-algèbre de dimension finie. Selon GEIGLE et LENZING, la catégorie perpendiculaire à droite MꞱ d'un A-module M est la sous-catégorie pleine de mod A définie par MꞱ = {NA |HomA(M, N) = 0 et Ext¹A (M, N) = 0}. Il a été démontré par GEIGLE et LENZING que si A est héréditaire et Ext¹A (M, M) = 0 alors MꞱ ≅ modB où B est une algèbre héréditaire de dimension finie telle que rgK0 (B) = rgK0 (A) - 1. De plus, par un résultat de GABRIEL, on a B ≅ KQ où Q est un carquois fini et acyclique. Le but de ce mémoire est de calculer la catégorie perpendiculaire d'un module indécomposable M (sur une algèbre héréditaire docile ou de représentation finie A) tel que Ext¹A(M, M) = 0, c'est-à-dire de calculer l'algèbre B telle que MꞱ ≅ modB.
Cited by 0
Mickaël Buchet, Emerson G. Escolar · Journal of Applied and Computational Topology · 2020
Abstract A recent work by Lesnick and Wright proposed a visualisation of 2D persistence modules by using their restrictions onto lines, giving a family of 1D persistence modules. We give a constructive proof that any 1D persistence module with finite support can be found as a restriction of some indecomposable 2D persistence module with finite support. As consequences of our construction, we are able to exhibit indecomposable 2D persistence modules whose support has holes as well as an indecomposable 2D persistence module containing all 1D persistence modules with finite support as line restrictions. Finally, we also show that any finite-rectangle-decomposable n D persistence module can be found as a restriction of some indecomposable $$(n+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> D persistence module.
Cited by 1Open access
Christopher C. Gill · arXiv (Cornell University) · 2012
We study the multiplicities of Young modules as direct summands of permutation modules on cosets of Young subgroups. Such multiplicities have become known as the p-Kostka numbers. We classify the indecomposable Young permutation modules, and, applying the Brauer construction for p-permutation modules, we give some new reductions for p-Kostka numbers. In particular we prove that p-Kostka numbers are preserved under multiplying partitions by p, and strengthen a known reduction given by Henke, corresponding to adding multiples of a p-power to the first row of a partition.
Cited by 3Open access
Mickaël Buchet, Emerson G. Escolar · arXiv (Cornell University) · 2019
A recent work by Lesnick and Wright proposed a visualisation of $2$D\npersistence modules by using their restrictions onto lines, giving a family of\n$1$D persistence modules. We give a constructive proof that any $1$D\npersistence module with finite support can be found as a restriction of some\nindecomposable $2$D persistence module with finite support. As consequences of\nour construction, we are able to exhibit indecomposable $2$D persistence\nmodules whose support has holes as well as an indecomposable $2$D persistence\nmodule containing all $1$D persistence modules with finite support as line\nrestrictions. Finally, we also show that any finite-rectangle-decomposable $n$D\npersistence module can be found as a restriction of some indecomposable\n$(n+1)$D persistence module.\n
Cited by 0Open access
Christopher C. Gill · arXiv (Cornell University) · 2012
We study the multiplicities of Young modules as direct summands of\npermutation modules on cosets of Young subgroups. Such multiplicities have\nbecome known as the p-Kostka numbers. We classify the indecomposable Young\npermutation modules, and, applying the Brauer construction for p-permutation\nmodules, we give some new reductions for p-Kostka numbers. In particular we\nprove that p-Kostka numbers are preserved under multiplying partitions by p,\nand strengthen a known reduction given by Henke, corresponding to adding\nmultiples of a p-power to the first row of a partition.\n
Cited by 0Open access
Christopher C. Gill · Journal of Algebra and Its Applications · 2013
We study the multiplicities of Young modules as direct summands of permutation modules on cosets of Young subgroups. Such multiplicities have become known as the p-Kostka numbers. We classify the indecomposable Young permutation modules, and, applying the Brauer construction for p-permutation modules, we give some new reductions for p-Kostka numbers. In particular, we prove that p-Kostka numbers are preserved under multiplying partitions by p, and strengthen a known reduction corresponding to adding multiples of a p-power to the first row of a partition.
Cited by 12
Wolfgang Hamernik · Journal of the London Mathematical Society · 1973
Cited by 5
David J. Benson, Markus Linckelmann · Journal of Pure and Applied Algebra · 2004
Cited by 5
Hagen Meltzer, Andrzej Skowroński · Journal of Algebra · 1986
Cited by 1
Christian Lomp · DergiPark (Istanbul University) · 2007
A module M is called hollow-lifting if every submodule N of M such that M/N is hollow contains a direct summand D \\subseteq N such that N/D is a small submodule of M/D. A module M is called lifting if such a direct summand D exists for every submodule N. We construct an indecomposable module M without non-zero hollow factor modules, showing that there are hollow-lifting modules which are not lifting. The existences of such modules had been left open in a recent work by N. Orhan, D. Keskin-Tütüncü and R. Tribak [2].
Cited by 2Open access
José Luis Gómez Pardo, Pedro A. Guil Asensio · Journal of Algebra · 2000
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Hiroyuki Tachikawa · Mathematische Zeitschrift · 1959
Cited by 50
Spencer E. Dickson, Kent R. Fuller · Pacific Journal of Mathematics · 1969
The algebras of the title are characterized as those having Tachikawa's strong left cyclic representation type (SLCRT) with the further property that every quotient of a principal indecomposable left module has square free socle. Moreover it turns out that these are precisely the algebras for which every indecomposable right module is quasi-injective in the sense of Johnson and Wong, and that these algebras have finite module type.
Cited by 84Open access
Masahisa Sato · Journal of Algebra · 1989
Cited by 0
Yukiyosi Kawada · Medical Entomology and Zoology · 1962
Cited by 3
Timothy J. Hodges, James Osterburg · Bulletin of the London Mathematical Society · 1987
Cited by 12
Zur Izhakian, Manfred Knebusch, Louis Rowen · arXiv (Cornell University) · 2015
We study quadratic forms on free modules with unique base, the situation that\narises in tropical algebra, and prove the analog of Witt's Cancellation\nTheorem. Also, the tensor product of an indecomposable bilinear module $(U,\n\\gamma)$ with an indecomposable quadratic module $(V,q) $ is indecomposable,\nwith the exception of one case, where two indecomposable components arise.\n
Cited by 1Open access
Louis Rowen, Zur Izhakian, Manfred Knebusch · Documenta Mathematica · 2016
We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancelation Theorem. Also, the tensor product of an indecomposable bilinear module (U, \gamma) with an indecomposable quadratic module (V,q) is indecomposable, with the exception of one case, where two indecomposable components arise.
Cited by 9Open access
A. A. Tuganbaev · Russian Mathematical Surveys · 1998
We assume that all rings are associative with a non-zero identity. We call a module strictly indecomposable if all of its quotient modules are indecomposable. We call a module a chain module if two of its submodules are congruent with respect to inclusion. We call a module uniform if any two of its non-zero submodules have a non-zero intersection. Any chain module is a strictly indecomposable uniform module. The group algebra of the direct product of two cyclic groups of order 2 over the field of two elements is a strictly indecomposable uniform non-chain module over itself. The principal results of this paper are Theorems 1 and 2: Theorem 1 generalizes analogous results from [1] for chain modules.
Cited by 1