On the structure of indecomposable modules over Artin algebras

In this paper we are going to use a result of H. Harada and Y. Sai concerning composition of nonisomorphisms between indecomposable modules and the theory of almost split sequences introduced in the representation theory of Artin algebras by M. Auslander and I. Reiten to obtain the inductive step in the second Brauer-Thrall conjecture.Section 1 is devoted to giving the necessary background in the theory of almost split sequences.As an application we get the first Brauer-Thrall conjecture for Artin algebras. This conjecture says that there is no bound on the length of the finitely generated indecomposable modules over an Artin algebra of infinite type, i.e., an Artin algebra such that there are infinitely many nonisomorphic indecomposable finitely generated modules. This result was first proved by A. V. Roiter [8] and later in general for Artin rings by M. Auslander [2] using categorical methods.

Download Full-text

Minimal representation-infinite artin algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100072546 ◽

1994 ◽

Vol 116 (2) ◽

pp. 229-243 ◽

Cited By ~ 23

Author(s):

Andrzej Skowroński

Keyword(s):

Jacobson Radical ◽

Minimal Representation ◽

Finitely Generated ◽

Artin Algebra ◽

Indecomposable Modules ◽

Artin Ring ◽

Artin Algebras ◽

Infinite Power

Let A be an artin algebra over a commutative artin ring R, mod A be the category of finitely generated right A-modules, and rad∞ (modA) be the infinite power of the Jacobson radical rad(modA) of modA. Recall that A is said to be representation-finite if mod A admits only finitely many non-isomorphic indecomposable modules. It is known that A is representation-finite if and only if rad∞ (mod A) = 0. Moreover, from the validity of the First Brauer–Thrall Conjecture [26, 2] we know that A is representation-finite if and only if there is a common bound on the length of indecomposable modules in mod A.

Download Full-text

Composition of irreducible morphisms in quasi-tubes

Journal of Algebra and Its Applications ◽

10.1142/s0219498817500712 ◽

2017 ◽

Vol 16 (04) ◽

pp. 1750071 ◽

Cited By ~ 2

Author(s):

Claudia Chaio ◽

Piotr Malicki

Keyword(s):

Module Category ◽

Indecomposable Modules ◽

Artin Algebras ◽

Irreducible Morphisms

We study the composition of irreducible morphisms between indecomposable modules lying in quasi-tubes of the Auslander–Reiten quivers of artin algebras in relation with the powers of the radical of their module category.

Download Full-text

Indecomposable modules over pure semisimple hereditary rings

Journal of Algebra ◽

10.1016/j.jalgebra.2012.09.004 ◽

2012 ◽

Vol 371 ◽

pp. 577-595 ◽

Cited By ~ 3

Author(s):

Nguyen Viet Dung ◽

José Luis García

Keyword(s):

Indecomposable Modules

Download Full-text

Indecomposable modules over one-dimensional Noetherian rings

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2006.03.007 ◽

2007 ◽

Vol 208 (2) ◽

pp. 739-760 ◽

Cited By ~ 4

Author(s):

Meral Arnavut ◽

Melissa Luckas ◽

Sylvia Wiegand

Keyword(s):

Noetherian Rings ◽

Indecomposable Modules ◽

One Dimensional

Download Full-text

Completely Indecomposable Modules

Canadian Journal of Mathematics ◽

10.4153/cjm-1949-013-3 ◽

1949 ◽

Vol 1 (2) ◽

pp. 125-152 ◽

Cited By ~ 4

Author(s):

Ernst Snapper

Keyword(s):

Abelian Group ◽

Commutative Ring ◽

Chain Condition ◽

Indecomposable Module ◽

Unit Element ◽

Indecomposable Modules ◽

Operator Domain ◽

Unit Operator ◽

Ascending Chain Condition

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.1-1. A is a commutative ring and has a unit element which is unit operator for .1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)

Download Full-text