Essential extensions of a direct sum of simple modules

2006 ◽  
pp. 15-23 ◽  
Author(s):  
K. I. Beidar ◽  
S. K. Jain ◽  
Ashish K. Srivastava
Keyword(s):  
Simple Modules ◽  
Direct Sum ◽  
Essential Extensions

When is every module with essential socle a direct sum of automorphism-invariant modules?

2019 ◽  
Vol 19 (10) ◽  
pp. 2050185
Author(s):  
Shahabaddin Ebrahimi Atani ◽  
Saboura Dolati Pish Hesari ◽  
Mehdi Khoramdel
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Essential Extension ◽  
Principal Ideal Ring ◽  
Von Neumann ◽  
Simple Modules ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Ideal Ring ◽  
Von Neumann Regular

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

On infinite direct sums of lifting modules

2017 ◽  
Vol 10 (03) ◽  
pp. 1750049
Author(s):  
M. Tamer Koşan ◽  
Truong Cong Quynh
Keyword(s):  
Present Article ◽  
Essential Extension ◽  
Injective Hull ◽  
Direct Sums ◽  
Simple Modules ◽  
Direct Sum

The aim of the present article is to investigate the structure of rings [Formula: see text] satisfying the condition: for any family [Formula: see text] of simple right [Formula: see text]-modules, every essential extension of [Formula: see text] is a direct sum of lifting modules, where [Formula: see text] denotes the injective hull. We show that every essential extension of [Formula: see text] is a direct sum of lifting modules if and only if [Formula: see text] is right Noetherian and [Formula: see text] is hollow. Assume that [Formula: see text] is an injective right [Formula: see text]-module with essential socle. We also prove that if every essential extension of [Formula: see text] is a direct sum of lifting modules, then [Formula: see text] is [Formula: see text]-injective. As a consequence of this observation, we show that [Formula: see text] is a right V-ring and every essential extension of [Formula: see text] is a direct sum of lifting modules for all simple modules [Formula: see text] if and only if [Formula: see text] is a right [Formula: see text]-V-ring.

On a class of separable modules

2021 ◽  
pp. 2250034
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Finitely Generated ◽  
Direct Sums ◽  
Simple Modules ◽  
Direct Sum ◽  
The Stability

A module [Formula: see text] is called [Formula: see text]-separable if every proper finitely generated submodule of [Formula: see text] is contained in a proper finitely generated direct summand of [Formula: see text]. Indecomposable [Formula: see text]-separable modules are shown to be exactly the simple modules. While direct summands of an [Formula: see text]-separable module do not inherit the property, in general, the question of the stability under direct sums is unanswered. But we obtain some partial answers. It is shown that any infinite direct sum of [Formula: see text]-separable modules is [Formula: see text]-separable. Also, we prove that if [Formula: see text] and [Formula: see text] are [Formula: see text]-separable modules such that [Formula: see text] is [Formula: see text]-projective, then [Formula: see text] is [Formula: see text]-separable. We conclude the paper by providing some characterizations of several classes of rings in terms of [Formula: see text]-separable modules. Among others, we prove that the class of rings [Formula: see text] for which every (injective) [Formula: see text]-module is [Formula: see text]-separable is exactly that of semisimple rings.

A NOTE ON ESSENTIAL EXTENSIONS OF SEMI-SIMPLE MODULES

2008 ◽  
Vol 07 (02) ◽  
pp. 225-230 ◽  
Author(s):  
NOYAN ER
Keyword(s):  
Chain Condition ◽  
Direct Sums ◽  
Simple Modules ◽  
Essential Extensions ◽  
Ascending Chain Condition

In a series of recent papers, Beidar, Jain and Srivastava studied the question as to when a ring R with the property that essential extensions of semi-simple right R-modules are direct sums of quasi-injectives is right Noetherian. Beidar and Jain proved that it is, when R is commutative or right q.f.d. In this note we extend their results proving the following: A ring R with this property is right Noetherian iff for some n ∈ ℕ, R/socn(RR) has ascending chain condition on essential non-two-sided right ideals (in particular, when R/socn(RR) is right q.f.d. or commutative). Also shown is the following: A ring is a right Noetherian right V-ring iff modules with essential socle are quasi-continuous/quasi-injective.

scholarly journals MONOLITHIC MODULES OVER NOETHERIAN RINGS

2011 ◽  
Vol 53 (3) ◽  
pp. 683-692 ◽  
Author(s):  
PAULA A. A. B. CARVALHO ◽  
IAN M. MUSSON
Keyword(s):  
Weyl Algebra ◽  
Noetherian Rings ◽  
Quantum Plane ◽  
Finiteness Conditions ◽  
Glasgow Math ◽  
Simple Modules ◽  
Injective Modules ◽  
Essential Extensions

AbstractWe study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantised Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained by Carvalho et al. (Carvalho et al., Injective modules over down-up algebras, Glasgow Math. J. 52A (2010), 53–59) for down-up algebras.

On Finite Essential Extensions of Torsion Free Abelian Groups

1996 ◽  
Vol 48 (5) ◽  
pp. 918-929 ◽  
Author(s):  
K. Benabdallah ◽  
M. A. Ouldbeddi
Keyword(s):  
Abelian Group ◽  
Finite Index ◽  
Free Abelian Group ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Rank One ◽  
Free Abelian Groups ◽  
Decomposable Groups ◽  
Essential Extensions ◽  
Torsion Free

AbstractLet A be a torsion free abelian group. We say that a group K is a finite essential extension of A if K contains an essential subgroup of finite index which is isomorphic to A. Such K admits a representation as (A ℤ xkx)/ℤky where y = Nx + a for some k x k matrix N over Z and α ∈ Ak satisfying certain conditions of relative primeness and ℤk = {(α1,..., αk) : αi, ∈ ℤ}. The concept of absolute width of an f.e.e. K of A is defined and it is shown to be strictly smaller than the rank of A. A kind of basis substitution with respect to Smith diagonal matrices is shown to hold for homogeneous completely decomposable groups. This result together with general properties of our representations are used to provide a self contained proof that acd groups with two critical types are direct sum of groups of rank one and two.

scholarly journals Abelian endoregular modules

2019 ◽  
Vol 19 (11) ◽  
pp. 2050202
Author(s):  
Mauricio Medina-Bárcenas ◽  
Hanna Sim
Keyword(s):  
Injective Hull ◽  
Endomorphism Rings ◽  
Von Neumann ◽  
Simple Modules ◽  
Direct Sum ◽  
Von Neumann Regular ◽  
Subdirect Products

In this paper, we introduce the notion of abelian endoregular modules as those modules whose endomorphism rings are abelian von Neumann regular. We characterize an abelian endoregular module [Formula: see text] in terms of its [Formula: see text]-generated submodules. We prove that if [Formula: see text] is an abelian endoregular module then so is every [Formula: see text]-generated submodule of [Formula: see text]. Also, the case when the (quasi-)injective hull of a module as well as the case when a direct sum of modules is abelian endoregular are presented. At the end, we study abelian endoregular modules as subdirect products of simple modules.

scholarly journals Abelian group in a topos of sheaves: torsion and essential extensions

1989 ◽  
Vol 12 (1) ◽  
pp. 89-98
Author(s):  
Kiran R. Bhutani
Keyword(s):  
Abelian Group ◽  
Hausdorff Space ◽  
Torsion Group ◽  
Torsion Subgroup ◽  
Direct Sum ◽  
Group A ◽  
Torsion Groups ◽  
Essential Extensions

We investigate the properties of torsion groups and their essential extensions in the category AbShL of Abellan groups in a topos of sheaves on a locale. We show that every torsion group is a direct sum of its p-primary components and for a torsion group A, the group [A,B] is reduced for any BεAbShL.. We give an example to show that in AbShL the torsion subgroup of an injective group need not be injective. Further we prove that if the locale is Boolean or finite then essential extensions of torsion groups are torsion. Finally we show that for a first countable hausdorff space X essential extensions of torsion groups in AbSh0(X) are torsion iff X is discrete.

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