scholarly journals 2-Visible Submodules and Fully 2-Visible Modules

2020 ◽  
pp. 24-28
Author(s):  
Mahmood S. Fiadh ◽  
Wafaa H. Hanoon
Keyword(s):  
Commutative Ring ◽  
Nonzero Ideal ◽  
The Relationship

Let X be a T -module, T is a commutative ring with identity and K be a proper submodule of X. In this paper we introduce the concepts of 2-visible submodules and fully 2-visible modules as a generalizations of visible submodules and fully visible modules resp., where K is said to be 2-visible whenever K=I2 K for every nonzero ideal I of T and AT -module X is called fully 2-visible if for any proper submodule of it is 2-visible.Study some of the properties of these concepts also discuss the relationship 2-visible submodules and fully 2-visible modules with 2-pure submoules and other related submodules and modules resp. are given

2-Visible Submodules and Fully 2-Visible Modules

2019 ◽  
pp. 1-6
Author(s):  
Mahmood S. Fiadh ◽  
Wafaa H. Hanoon
Keyword(s):  
Commutative Ring ◽  
Nonzero Ideal ◽  
The Relationship

Let be a -module, T is a commutative ring with identity and be a proper submodule of . In this paper we introduce the concepts of 2-visible submodules and fully 2-visible modules as a generalizations of visible submodules and fully visible modules resp., where is said to be 2-visible whenever for every nonzero ideal of and A -module is called fully 2-visible if for any proper submodule of it is 2-visible.Study some of the properties of these concepts also discuss the relationship 2-visible submodules and fully 2-visible modules with 2-pure submoules and other related submodules and modules resp. are given.

Baer-invariants and extensions relative to a variety

1967 ◽  
Vol 63 (3) ◽  
pp. 569-578 ◽  
Author(s):  
Abraham S.-T. Lue
Keyword(s):  
Commutative Ring ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Jordan Algebras ◽  
Associative Algebras ◽  
Definition Of ◽  
The Relationship

This paper examines the relationship between extensions in a variety and general extensions in the category of associative algebras. Our associative algebras are all unitary, over some fixed commutative ring Λ with identity, but while our discussion will be restricted to this category, it is clear that obvious analogues exist for groups, Lie algebras and Jordan algebras. (We use the notion of a bimultiplication of an associative algebra. In (2), Knopfmacher gives the definition of a bimultiplication in any variety of linear algebras.)

The Intersection Property of Annihilators of Both Modules and Maximal Submodules Belonging to the Same Module

2020 ◽  
Vol 17 (2) ◽  
pp. 552-555
Author(s):  
Hatam Yahya Khalaf ◽  
Buthyna Nijad Shihab
Keyword(s):  
Commutative Ring ◽  
Intersection Property ◽  
Direct Sum ◽  
Unitary Module ◽  
Maximal Submodule ◽  
The Relationship

During that article T stands for a commutative ring with identity and that S stands for a unitary module over T. The intersection property of annihilatoers of a module X on a ring T and a maximal submodule W of M has been reviewed under this article where he provide several examples that explain that the property. Add to this a number of equivalent statements about the intersection property have been demonstrated as well as the direct sum of module that realize that the characteristic has studied here we proved that the modules that achieve the intersection property are closed under the direct sum with a specific condition. In addition to all this, the relationship between the modules that achieve the above characteristics with other types of modules has been given.

scholarly journals Diameter and girth of Torsion Graph

2014 ◽  
Vol 22 (3) ◽  
pp. 127-136
Author(s):  
P. Malakooti Rad ◽  
S. Yassemi ◽  
Sh. Ghalandarzadeh ◽  
P. Safari
Keyword(s):  
Commutative Ring ◽  
The Relationship ◽  
Nonzero Torsion

AbstractLet R be a commutative ring with identity. Let M be an R-module and T (M)* be the set of nonzero torsion elements. The set T(M)* makes up the vertices of the corresponding torsion graph, ΓR(M), with two distinct vertices x, y ∈ T(M)* forming an edge if Ann(x) ∩ Ann(y) ≠ 0. In this paper we study the case where the graph ΓR(M) is connected with diam(ΓR(M)) ≤ 3 and we investigate the relationship between the diameters of ΓR(M) and ΓR(R). Also we study girth of ΓR(M), it is shown that if ΓR(M) contains a cycle, then gr(ΓR(M)) = 3.

Ideal-Based k-Zero-Divisor Hypergraph of Commutative Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 655-672
Author(s):  
K. Selvakumar ◽  
M. Subajini
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Zero Divisor ◽  
Necessary Conditions ◽  
Commutative Rings ◽  
Nonzero Ideal ◽  
Fixed Integer ◽  
Zero Divisors ◽  
Vertex Set ◽  
The Ideal

Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] a fixed integer. The ideal-based [Formula: see text]-zero-divisor hypergraph [Formula: see text] of [Formula: see text] has vertex set [Formula: see text], the set of all ideal-based [Formula: see text]-zero-divisors of [Formula: see text], and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge in [Formula: see text] if and only if [Formula: see text] and the product of the elements of any [Formula: see text]-subset of [Formula: see text] is not in [Formula: see text]. In this paper, we show that [Formula: see text] is connected with diameter at most 4 provided that [Formula: see text] for all ideal-based 3-zero-divisor hypergraphs. Moreover, we find the chromatic number of [Formula: see text] when [Formula: see text] is a product of finite fields. Finally, we find some necessary conditions for a finite ring [Formula: see text] and a nonzero ideal [Formula: see text] of [Formula: see text] to have [Formula: see text] planar.

Derivations in Prime Rings

1983 ◽  
Vol 26 (3) ◽  
pp. 267-270 ◽  
Author(s):  
Jeffrey Bergen
Keyword(s):  
Commutative Ring ◽  
Prime Ring ◽  
Simple Algebra ◽  
Finitely Generated ◽  
Prime Rings ◽  
Finite Dimensional ◽  
The Relationship

AbstractLet R be a prime ring and d≠0 a derivation of R. We examine the relationship between the structure of R and that of d(R). We prove that if R is an algebra over a commutative ring A such that d(R) is a finitely generated submodule then R is an order in a simple algebra finite dimensional over its center.

Envelopes, covers and semidualizing modules

2019 ◽  
Vol 18 (07) ◽  
pp. 1950137
Author(s):  
Lixin Mao
Keyword(s):  
Commutative Ring ◽  
Noetherian Rings ◽  
Artinian Rings ◽  
Injective Modules ◽  
Semidualizing Modules ◽  
Coherent Rings ◽  
The Relationship

Given an [Formula: see text]-module [Formula: see text] and a class of [Formula: see text]-modules [Formula: see text] over a commutative ring [Formula: see text], we investigate the relationship between the existence of [Formula: see text]-envelopes (respectively, [Formula: see text]-covers) and the existence of [Formula: see text]-envelopes or [Formula: see text]-envelopes (respectively, [Formula: see text]-covers or [Formula: see text]-covers) of modules. As a consequence, we characterize coherent rings, Noetherian rings, perfect rings and Artinian rings in terms of envelopes and covers by [Formula: see text]-projective, [Formula: see text]-flat, [Formula: see text]-injective and [Formula: see text]-[Formula: see text]-injective modules, where [Formula: see text] is a semidualizing [Formula: see text]-module.

scholarly journals COMODULES AND CONTRAMODULES

2010 ◽  
Vol 52 (A) ◽  
pp. 151-162
Author(s):  
ROBERT WISBAUER
Keyword(s):  
Tensor Product ◽  
Commutative Ring ◽  
The Right ◽  
The Relationship

AbstractAlgebras A and coalgebras C over a commutative ring R are defined by properties of the (endo)functors A ⊗R – and C ⊗R – on the category of R-modules R. Generalising these notions, monads and comonads were introduced on arbitrary categories, and it turned out that some of their basic relations do not depend on the specific properties of the tensor product. In particular, the adjoint of any comonad is a monad (and vice versa), and hence, for any coalgebra C, HomR(C, –), the right adjoint of C ⊗R –, is a monad on R. The modules for the monad HomR(C, –) were called contramodules by Eilenberg–Moore and the purpose of this talk is to outline the related constructions and explain the relationship between C-comodules and C-contramodules. The results presented grew out from cooperation with G. Böhm, T. Brzeziński and B. Mesablishvili.

ALMOST PERFECT LOCAL DOMAINS AND THEIR DOMINATING ARCHIMEDEAN VALUATION DOMAINS

2002 ◽  
Vol 01 (04) ◽  
pp. 451-467 ◽  
Author(s):  
PAOLO ZANARDO
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Integral Closure ◽  
Nonzero Ideal ◽  
Valuation Domain ◽  
Local Domain ◽  
Valuation Domains ◽  
Domain V

A commutative ring R is said to be almost perfect if R/I is perfect for every nonzero ideal I of R. We prove that an almost perfect local domain R is dominated by a unique archimedean valuation domain V of its field of quotients Q if and only if the integral closure of R contains an ideal of V. We show how to construct almost perfect local domains dominated by finitely many archimedean valuation domains. We provide several examples illustrating various possible situations. In particular, we construct an almost perfect local domain whose maximal ideal is not almost nilpotent.

SOME RESULTS ON THE COMPLEMENT OF THE ZERO-DIVISOR GRAPH OF A COMMUTATIVE RING

2011 ◽  
Vol 10 (03) ◽  
pp. 573-595 ◽  
Author(s):  
S. VISWESWARAN
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
The Relationship

Let R be a commutative ring with identity admitting at least two nonzero zero-divisors. First, in this article we determine when the complement of the zero-divisor graph of R is connected and also determine its diameter when it is connected. Second, in this article we study the relationship between the connectedness of the complement of the zero-divisor graph of R to that of the connectedness of the complement of the zero-divisor graph of T where either T = R[x] or T = R[[x]] and we study the relationship between their diameters in the case when both the graphs are connected. Finally, we give some examples to illustrate some of the results proved in this article.

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