On the composite of two irreducible morphisms in radical cube (original) (raw)

On the composite of three irreducible morphisms over string algebras

The São Paulo Journal of Mathematical Sciences, 2010

We characterize the representation-finite triangular string algebras having a path of irreducible morphisms of length three between pairwise non-isomorphic modules whose composite lies in the fourth power of the radical.

The composite of irreducible morphisms in regular components

Colloquium Mathematicum, 2011

We study when the composite of n irreducible morphisms between modules in a regular component of the Auslander-Reiten quiver is non-zero and lies in the n + 1-th power of the radical of the module category. We prove that in this case such a composite belongs to ∞. We apply these results to characterize those string algebras having n irreducible morphisms between band modules such that their composite is a non-zero morphism in n+1 .

Representations of right 3-Nakayama algebras

arXiv (Cornell University), 2018

In this paper we study the category of finitely generated modules over a right 3-Nakayama artin algebra. First we give a characterization of right 3-Nakayama artin algebras and then we give a complete list of non-isomorphic finitely generated indecomposable modules over any right 3-Nakayama artin algebra. Also we compute all almost split sequences for the class of right 3-Nakayama artin algebras. Finally, we classify finite dimensional right 3-Nakayama algebras in terms of their quivers with relations. 1. introduction Let R be a commutative artinian ring. An R-algebra Λ is called an artin algebra if Λ is finitely generated as a R-module. Let Λ be an artin algebra. A right Λ-module M is called uniserial (1-factor serial) if it has a unique composition series. An artin algebra Λ is called Nakayama algebra if any indecomposable right Λ-module is uniserial. The class of Nakayama algebras is one the important class of representation finite algebras whose representation theory completely understood [3]. According to [5, Definition 2.1], a non-uniserial right Λ-module M of length l is called n-factor serial (l ≥ n > 1), if M rad l−n (M) is uniserial and M rad l−n+1 (M) is not uniserial. An artin algebra Λ is called right n-Nakayama if every indecomposable right Λ-module is i-factor serial for some 1 ≤ i ≤ n and there exists at least one indecomposable n-factor serial right Λ-module [5, Definition 2.2]. The authors in [5] showed that the class of right n-Nakayama algebras provide a nice partition of the class of representation finite artin algebras. More precisely, the authors proved that an artin algebra Λ is representation finite if and only if Λ is right n-Nakayama for some positive integer n [5, Theorem 2.18]. The first part of this partition is the class of Nakayama algebras and the second part is the class of right 2-Nakayama algebras. Indecomposable modules and almost split sequences for the class of right 2-Nakayama algebras are classified in section 5 of [5]. In this paper we will study the class of right 3-Nakayama algebras. We first show that an artin algebra Λ which is neither Nakayama nor right 2-Nakayama is right 3-Nakayama if and only if every indecomposable right Λ-module of length greater than 4 is uniserial and every indecomposable right Λmodule of length 4 is local. Then we classify all indecomposable modules and almost split sequences over a right 3-Nakayama artin algebra. We also show that finite dimensional right 3-Nakayama algebras are special biserial and we describe all finite dimensional right 3-Nakayama algebras by their quivers and relations. Riedtmann in [6] and [7], by using the covering theory, classified representation-finite self-injective algebras.

The composite of irreducible morphisms in standard components

Journal of Algebra, 2010

In this work, we consider standard components of the Auslander-Reiten quiver with trivial valuation. We give a characterization of when there are n irreducible morphisms between modules in such a component with non-zero composite belonging to the n + 1-th power of the radical. We prove that a necessary condition for their existence is that it has to be a non-zero cycle or a non-zero bypass in the component. For directed algebras, we prove that the composite of n irreducible morphisms between indecomposable modules belongs to a greater power of the radical, greater than n, if and only if it is zero.

On algebras of finite representation type

Transactions of The American Mathematical Society, 1969

Introduction. Since D. G. Higman proved that bounded representation type and finite representation type are equivalent for group algebras at prime characteristic, there has been a renewed interest in the Brauer-Thrall conjecture that bounded representation type implies finite representation type for arbitrary algebras. The main purpose of this paper is to present a new approach to this conjecture by showing the relevance (when the base field is algebraically closed) of questions concerning the structure of indecomposable modules of certain special types, namely, the stable (every maximal submodule is indecomposable), the costable (having the dual property), and the stable-costable (having both properties) indecomposable modules. The main tools are the Sandwich Lemma (1.2) which is proved using an old observation of É. Goursat, an observation of A. Heller, C. W. Curtis, and D. Zelinsky concerning quasifrobenius (QF) rings (Proposition 2.1), and a general interlacing technique similar to methods used by Jans, Tachikawa, and Colby for building up large indecomposable modules of finite length which has validity in any abelian category (Theorem 3.1).

On cycles of length three

2021

We prove that if A is a string algebra then there are not three irreducible morphisms between indecomposable A-modules such that its composition belongs to R\R, whenever the compositions of two of them are not in R. Moreover, for any positive integer n ≥ 3, we show that there are n irreducible morphisms such that their composition is in R\R.

Degrees of irreducible morphisms in standard components

Journal of Pure and Applied Algebra, 2010

We study the left degree of an irreducible morphism f : X → r i=1 Y i with X and Y i indecomposable modules in a standard component of the Auslander-Reiten quiver, for 1 ≤ i ≤ r. Two criteria to determine whether the left degree of these irreducible morphisms is finite or infinite are given, for standard algebras. We also study which of them has left degree two.