scholarly journals On α-almost Artinian modules

2016 ◽  
Vol 14 (1) ◽  
pp. 404-413 ◽  
Author(s):  
Maryam Davoudian ◽  
Ahmad Halali ◽  
Nasrin Shirali
Keyword(s):  
Homomorphic Image ◽  
Krull Dimension ◽  
Artinian Modules ◽  
Artinian Module

AbstractIn this article we introduce and study the concept of α-almost Artinian modules. We show that each α-almost Artinian module M is almost Artinian (i.e., every proper homomorphic image of M is Artinian), where α ∈ {0,1}. Using this concept we extend some of the basic results of almost Artinian modules to α-almost Artinian modules. Moreover we introduce and study the concept of α-Krull modules. We observe that if M is an α-Krull module then the Krull dimension of M is either α or α + 1.

scholarly journals Co-Cohen-Macaulay Artinian modules over commutative rings

1996 ◽  
Vol 38 (3) ◽  
pp. 359-366 ◽  
Author(s):  
I. H. Denizler ◽  
R. Y. Sharp
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Krull Dimension ◽  
Artinian Modules ◽  
Artinian Module

In [7], Z. Tang and H. Zakeri introduced the concept of co-Cohen-Macaulay Artinian module over a quasi-local commutative ring R (with identity): a non-zero Artinian R-module A is said to be a co-Cohen-Macaulay module if and only if codepth A = dim A, where codepth A is the length of a maximalA-cosequence and dimA is the Krull dimension of A as defined by R. N. Roberts in [2]. Tang and Zakeriobtained several properties of co-Cohen-Macaulay Artinian R-modules, including a characterization of such modules by means of the modules of generalized fractions introduced by Zakeri and the present second author in [6]; this characterization is explained as follows.

Some properties of Artinian modules and applications

2018 ◽  
Vol 17 (02) ◽  
pp. 1850019
Author(s):  
Tran Nguyen An
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Vanishing Theorem ◽  
Strong Connection ◽  
Artinian Modules ◽  
Local Cohomology Modules ◽  
Artinian Module ◽  
Base Ring ◽  
System Of Parameters ◽  
Cohomology Modules

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] be an Artinian [Formula: see text]-module. Consider the following property for [Formula: see text] : [Formula: see text] In this paper, we study the property (∗) of [Formula: see text] in order to investigate the relation of system of parameters between [Formula: see text] and the ring [Formula: see text]. We also show that the property (∗) of [Formula: see text] has strong connection with the structure of base ring. Some applications to cofinite Artinian module are given. These are generalizations of [N. Abazari and K. Bahmanpour, A note on the Artinian cofinite modules, Comm. Algebra. 42 (2014) 1270–1275; G. Ghasemi, K. Bahmanpour and J. Azami, On the cofiniteness of Artinian local cohomology modules, J. Algebra Appl. 15(4) (2016), Article ID: 1650070, 8 pp.] A generalization of Lichtenbaum–Hartshorne Vanishing Theorem is also given in this paper.

On Sequentially Co-Cohen–Macaulay Modules

2007 ◽  
Vol 14 (03) ◽  
pp. 455-468
Author(s):  
Nguyen Thi Dung
Keyword(s):  
Artinian Modules ◽  
Local Homology ◽  
Matlis Duality ◽  
Artinian Module

In this paper, we define the notion of dimension filtration of an Artinian module and study a class of Artinian modules, called sequentially co-Cohen–Macaulay modules, which contains strictly all co-Cohen–Macaulay modules. Some characterizations of co-Cohen–Macaulayness in terms of the Matlis duality and of local homology are also given.

Artinian modules over commutative rings

1992 ◽  
Vol 111 (1) ◽  
pp. 25-33 ◽  
Author(s):  
R. Y. Sharp
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Artinian Modules ◽  
Noetherian Local Ring ◽  
Matlis Duality ◽  
Artinian Module

In 5, I provided a method whereby the study of an Artinian module A over a commutative ring R (throughout the paper, R will denote a commutative ring with identity) can, for some purposes at least, be reduced to the study of an Artinian module A' over a complete (Noetherian) local ring; in the latter situation, Matlis' duality 1 (alternatively, see 6, ch. 5) is available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

Hilbert–Samuel multiplicity and Northcott’s inequality relative to an Artinian module

2019 ◽  
Vol 30 (02) ◽  
pp. 379-396
Author(s):  
V. H. Jorge Pérez ◽  
T. H. Freitas
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Artinian Modules ◽  
Dimension Formula ◽  
Hilbert Coefficients ◽  
Artinian Module ◽  
Samuel Multiplicity

Let [Formula: see text] be a commutative quasi-local ring (with identity [Formula: see text]), and let [Formula: see text] be an [Formula: see text]-ideal such that [Formula: see text]. For [Formula: see text] an Artinian [Formula: see text]-module of N-dimension [Formula: see text], we introduce the notion of Hilbert-coefficients of [Formula: see text] relative to [Formula: see text] and give several properties. When [Formula: see text] is a co-Cohen–Macaulay [Formula: see text]-module, we establish the Northcott’s inequality for Artinian modules. As applications, we show some formulas involving the Hilbert coefficients and we investigate the behavior of these multiplicities when the module is the local cohomology module.

Automorphism-invariant semi-Artinian modules

2017 ◽  
Vol 16 (02) ◽  
pp. 1750029 ◽  
Author(s):  
A. A. Tuganbaev
Keyword(s):  
Artinian Modules ◽  
Artinian Module

It is proved that semi-Artinian module [Formula: see text] is an automorphism-invariant module if and only if [Formula: see text] is an automorphism-extendable module.

scholarly journals A note on reductions of ideals relative to an Artinian module

1993 ◽  
Vol 35 (2) ◽  
pp. 219-224 ◽  
Author(s):  
A.-J. Taherizadeh
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Integral Closure ◽  
Artinian Modules ◽  
Artinian Module ◽  
Image Position ◽  
Positive Integers

The concept of reduction and integral closure of ideals relative to Artinian modules were introduced in [7]; and we summarize some of the main aspects now.Let A be a commutative ring (with non-zero identity) and let a, b be ideals of A. Suppose that M is an Artinian module over A. We say that a is a reduction of b relative to M if a ⊆ b and there is a positive integer s such that)O:Mabs)=(O:Mbs+l).An element x of A is said to be integrally dependent on a relative to M if there exists n y ℕ(where ℕ denotes the set of positive integers) such thatIt is shown that this is the case if and only if a is a reduction of a+Ax relative to M; moreoverᾱ={x ɛ A: xis integrally dependent on a relative to M}is an ideal of A called the integral closure of a relative to M and is the unique maximal member of℘ = {b: b is an ideal of A which has a as a reduction relative to M}.

scholarly journals The nice m-system of parameters for Artinian modules

2020 ◽  
pp. 158-165
Keyword(s):  
Artinian Modules ◽  
Local Homology ◽  
Artinian Module ◽  
Systems Of Parameters ◽  
Definition Of ◽  
System Of Parameters

This paper restates the definition of the nice m-system of parameters for Artinian modules. It also shows its effects on the differences between lengths and multiplicities of certain systems of parameters for Artinian modules: In particular, if is a nice m-system of parameters then the function is a polynomial having very nice form. Moreover, we will prove some properties of the nice m-system of parameters for Artinian modules. Especially, its effect on the annihilation of local homology modules of Artinian module A.

Modules with epimorphisms on chains of submodules

2017 ◽  
Vol 16 (06) ◽  
pp. 1750101 ◽  
Author(s):  
R. Dastanpour ◽  
A. Ghorbani
Keyword(s):  
Finite Number ◽  
Homomorphic Image ◽  
Principal Ideal ◽  
Direct Sums ◽  
Matrix Rings ◽  
Principal Ideal Ring ◽  
Artinian Modules ◽  
Semisimple Modules ◽  
Ideal Ring ◽  
Principal Ideal Domains

We say that an [Formula: see text]-module [Formula: see text] satisfies epi-ACC on submodules if in every ascending chain of submodules of [Formula: see text], except probably a finite number, each module in chain is a homomorphic image of the next one. Noetherian modules, semisimple modules and Prüfer [Formula: see text]-groups have this property. Direct sums of modules with epi-ACC on submodules need not have this property. If [Formula: see text] satisfies epi-ACC on submodules, then [Formula: see text] is quasi-Frobenius. As a consequence, a ring [Formula: see text] in which all modules satisfy epi-ACC on submodules is an artinian principal ideal ring. Dually, we say that an [Formula: see text]-module [Formula: see text] satisfies epi-DCC on submodules if in every descending chain of submodules of [Formula: see text], except probably a finite number, each module in chain is a homomorphic image of the preceding. Artinian modules, semisimple modules and free modules over commutative principal ideal domains are examples of such modules. A semiprime right Goldie ring satisfies epi-DCC on right ideals if and only if it is a finite product of full matrix rings over principal right ideal domains. A ring [Formula: see text] for which all modules satisfy epi-DCC on submodules must be an artinian principal ideal ring.

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