scholarly journals The Relationships between Relatively Cancellation Modules and Certain Types of Modules

2011 ◽  
Vol 8 (1) ◽  
pp. 183-187
Author(s):  
Baghdad Science Journal
Keyword(s):  
Commutative Ring ◽  
Artinian Modules ◽  
Pure Submodules

Let R be a commutative ring with identity and M be unitary (left) R-module. The principal aim of this paper is to study the relationships between relatively cancellation module and multiplication modules, pure submodules and Noetherian (Artinian) modules.

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.

scholarly journals BILINEAR MAPS ON ARTINIAN MODULES

2012 ◽  
Vol 11 (05) ◽  
pp. 1250090
Author(s):  
GEORGE M. BERGMAN
Keyword(s):  
Commutative Ring ◽  
Finite Length ◽  
Nonzero Element ◽  
Bilinear Maps ◽  
Bilinear Map ◽  
Noncommutative Rings ◽  
Artinian Modules

It is shown that if a bilinear map f : A × B → C of modules over a commutative ring k is nondegenerate (i.e. if no nonzero element of A annihilates all of B, and vice versa), and A and B are Artinian, then A and B are of finite length. Some consequences are noted. Counterexamples are given to some attempts to generalize the above result to balanced bilinear maps of bimodules over noncommutative rings, while the question is raised whether other such generalizations are true.

scholarly journals Some Results on Pure Submodules Relative to Submodule

2015 ◽  
Vol 12 (4) ◽  
pp. 833-837
Author(s):  
Baghdad Science Journal
Keyword(s):  
Commutative Ring ◽  
Extending Module ◽  
Closed Submodules ◽  
Pure Submodules

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be pure relative to submodule T of M (Simply T-pure) if for each ideal A of R, N?AM=AN+T?(N?AM). In this paper, the properties of the following concepts were studied: Pure essential submodules relative to submodule T of M (Simply T-pure essential),Pure closed submodules relative to submodule T of M (Simply T-pure closed) and relative pure complement submodule relative to submodule T of M (Simply T-pure complement) and T-purely extending. We prove that; Let M be a T-purely extending module and let N be a T-pure submodule of M. If M has the T-PIP, then N is T-purely extending.

scholarly journals Notes on Approximately Pure Submodules

2013 ◽  
Vol 10 (4) ◽  
pp. 1269-1272
Author(s):  
Baghdad Science Journal
Keyword(s):  
Commutative Ring ◽  
Intersection Property ◽  
Pure Submodules

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be approximately pure submodule of an R-module, if for each ideal I of R. The main purpose of this paper is to study the properties of the following concepts: approximately pure essentialsubmodules, approximately pure closedsubmodules and relative approximately pure complement submodules. We prove that: when an R-module M is an approximately purely extending modules and N be Ap-puresubmodulein M, if M has the Ap-pure intersection property then N is Ap purely extending.

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}.

On Modules Satisfying the Descending Chain Condition on Cyclic Submodules

2020 ◽  
Vol 27 (03) ◽  
pp. 531-544
Author(s):  
Farid Kourki ◽  
Rachid Tribak
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Chain Condition ◽  
Finitely Generated ◽  
Artinian Modules ◽  
Cyclic Submodules ◽  
Descending Chain Condition

A module satisfying the descending chain condition on cyclic submodules is called coperfect. The class of coperfect modules lies properly between the class of locally artinian modules and the class of semiartinian modules. Let R be a commutative ring with identity. We show that every semiartinian R-module is coperfect if and only if R is a T-ring. It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if 𝔪/𝔪2 is a finitely generated R-module for every maximal ideal 𝔪 of R.

A Cohen-type theorem for w-Artinian modules

2020 ◽  
pp. 2150106 ◽  
Author(s):  
Dechuan Zhou ◽  
Hwankoo Kim ◽  
Kui Hu
Keyword(s):  
Commutative Ring ◽  
Chinese Remainder Theorem ◽  
Type Theorem ◽  
Artinian Modules ◽  
Ideal Formula ◽  
Theoretic Version

Let [Formula: see text] be a commutative ring with identity. In this paper, a Cohen-type theorem for [Formula: see text]-Artinian modules is given, i.e. a [Formula: see text]-cofinitely generated [Formula: see text]-module [Formula: see text] is [Formula: see text]-Artinian if and only if [Formula: see text] is [Formula: see text]-cofinitely generated for every prime [Formula: see text]-ideal [Formula: see text] of [Formula: see text]. As a by-product of the proof, we also obtain a detailed representation of elements of a [Formula: see text]-module and the [Formula: see text]-theoretic version of the Chinese remainder theorem for both modules and rings.

scholarly journals Integral closures of ideals relative to Artinian modules, and exact sequences

1992 ◽  
Vol 34 (1) ◽  
pp. 103-107 ◽  
Author(s):  
R. Y. Sharp ◽  
Y. Tiraş
Keyword(s):  
Commutative Ring ◽  
Integral Closure ◽  
Artinian Modules ◽  
Integral Closures ◽  
Exact Sequences

In [3], Sharp and Taherizadeh introduced concepts of reduction and integral closure of an ideal I of a commutative ring R (with identity) relative to an Artinian R-module A, and they showed that these concepts have properties which reflect some of those of the classical concepts of reduction and integral closure introduced by Northcott and Rees in [2].

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.

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