On Algebras Stably Equivalent to an Hereditary Artin Algebra

Let Λ be an artin algebra, that is, an artin ring that is a finitely generated module over its center C which is also an artin ring. We denote by mod Λ the category of finitely generated left Λ-modules. We recall that the category of finitely generated modules modulo projectives is the category given by the following data: the objects are the finitely generated Λ-modules.

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Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

Algebra Colloquium ◽

10.1142/s1005386716000596 ◽

2016 ◽

Vol 23 (04) ◽

pp. 701-720 ◽

Cited By ~ 2

Author(s):

Xiangui Zhao ◽

Yang Zhang

Keyword(s):

Computational Method ◽

Finitely Generated ◽

Finitely Generated Module ◽

Polynomial Algebras ◽

Ore Extensions ◽

Basis Method ◽

Finitely Generated Modules ◽

Kirillov Dimension

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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ON A NEW INVARIANT OF FINITELY GENERATED MODULES OVER LOCAL RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498810004324 ◽

2010 ◽

Vol 09 (06) ◽

pp. 959-976 ◽

Cited By ~ 4

Author(s):

NGUYEN TU CUONG ◽

DOAN TRUNG CUONG ◽

HOANG LE TRUONG

Keyword(s):

Local Ring ◽

Negative Integer ◽

Local Rings ◽

Finitely Generated ◽

Finitely Generated Module ◽

Polynomial Type ◽

Systems Of Parameters ◽

Finitely Generated Modules

Let M be a finitely generated module on a local ring R and [Formula: see text] a filtration of submodules of M such that do < d1 < ⋯ < dt = d, where di = dim Mi. This paper is concerned with a non-negative integer [Formula: see text] which is defined as the least degree of all polynomials in n1, …, nd bounding above the function [Formula: see text] We prove that [Formula: see text] is independent of the choice of good systems of parameters [Formula: see text]. When [Formula: see text] is the dimension filtration of M, we can use the polynomial type of Mi/Mi-1 and the dimension of the non-sequentially Cohen–Macaulay locus of M to compute [Formula: see text], and also to study the behavior of it under local flat homomorphisms.

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The Inductive Step of the Second Brauer-Thrall Conjecture

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-026-0 ◽

1980 ◽

Vol 32 (2) ◽

pp. 342-349 ◽

Cited By ~ 9

Author(s):

Sverre O. Smalø

Keyword(s):

Representation Theory ◽

Finitely Generated ◽

Inductive Step ◽

Artin Algebra ◽

Indecomposable Modules ◽

Infinite Type ◽

Categorical Methods ◽

Artin Algebras ◽

Almost Split Sequences ◽

Finitely Generated Modules

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.

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Characteristic polynomials of finitely generated modules over Weyl algebras

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700022425 ◽

2000 ◽

Vol 61 (3) ◽

pp. 387-403 ◽

Cited By ~ 2

Author(s):

Alexander B. Levin

Keyword(s):

Characteristic Polynomial ◽

Gröbner Basis ◽

Weyl Algebra ◽

Characteristic Polynomials ◽

Grobner Basis ◽

Finitely Generated ◽

Finitely Generated Module ◽

Weyl Algebras ◽

Dimension Polynomial ◽

Finitely Generated Modules

In this paper we modify the classical Gröbner basis technique and prove the existence of a characteristic polynomial in two variables associated with a finitely generated module over a Weyl algebra. We determine invariants of such a polynomial and show that some of the invariants are not carried by the Bernstein dimension polynomial of the module.

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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.

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Injective stabilization of additive functors, III. Asymptotic stabilization of the tensor product

Algebra and Discrete Mathematics ◽

10.12958/adm1728 ◽

2021 ◽

Vol 31 (1) ◽

pp. 120-151

Author(s):

Alex Martsinkovsky ◽

◽

Jeremy Russell ◽

Keyword(s):

Tensor Product ◽

Iterative Procedure ◽

Finitely Generated ◽

Artin Algebra ◽

Finitely Generated Module ◽

Asymptotic Stabilization ◽

Stable Cohomology ◽

Derived Functors ◽

Finitely Presented ◽

Connected Sequence

The injective stabilization of the tensor product is subjected to an iterative procedure that utilizes its bifunctor property. The limit of this procedure, called the asymptotic stabilization of the tensor product, provides a homological counterpart of Buchweitz's asymptotic construction of stable cohomology. The resulting connected sequence of functors is isomorphic to Triulzi's J-completion of the Tor functor. A comparison map from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be always epic. The category of finitely presented functors is shown to be complete and cocomplete. As a consequence, the inert injective stabilization of the tensor product with fixed variable a finitely generated module over an artin algebra is shown to be finitely presented. Its defect and consequently all right-derived functors are determined. New notions of asymptotic torsion and cotorsion are introduced and are related to each other.

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On a Class of Automaton Algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091516000134 ◽

2016 ◽

Vol 60 (1) ◽

pp. 31-38 ◽

Cited By ~ 1

Author(s):

Ferran Cedó ◽

Jan Okniński

Keyword(s):

Finitely Generated ◽

Finitely Generated Module ◽

Commutative Subalgebra

AbstractWe show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

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Prime uniserial modules and rings

Journal of Algebra and Its Applications ◽

10.1142/s021949881950035x ◽

2019 ◽

Vol 18 (02) ◽

pp. 1950035 ◽

Cited By ~ 1

Author(s):

M. Behboodi ◽

Z. Fazelpour

Keyword(s):

Commutative Rings ◽

Finitely Generated ◽

Finitely Generated Module ◽

Direct Sum ◽

Dedekind Domains ◽

Prime Submodules ◽

Uniserial Modules ◽

Structure Formula

We define prime uniserial modules as a generalization of uniserial modules. We say that an [Formula: see text]-module [Formula: see text] is prime uniserial ([Formula: see text]-uniserial) if its prime submodules are linearly ordered by inclusion, and we say that [Formula: see text] is prime serial ([Formula: see text]-serial) if it is a direct sum of [Formula: see text]-uniserial modules. The goal of this paper is to study [Formula: see text]-serial modules over commutative rings. First, we study the structure [Formula: see text]-serial modules over almost perfect domains and then we determine the structure of [Formula: see text]-serial modules over Dedekind domains. Moreover, we discuss the following natural questions: “Which rings have the property that every module is [Formula: see text]-serial?” and “Which rings have the property that every finitely generated module is [Formula: see text]-serial?”.

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