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.

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.

PURE-INJECTIVITY FROM A DIFFERENT PERSPECTIVE

2017 ◽  
Vol 60 (1) ◽  
pp. 135-151 ◽  
Author(s):  
S. R. LÓPEZ-PERMOUTH ◽  
J. MASTROMATTEO ◽  
Y. TOLOOEI ◽  
B. UNGOR
Keyword(s):  
Point Of View ◽  
Endomorphism Rings ◽  
Von Neumann ◽  
Basic Properties ◽  
Injective Modules ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Pure Extension ◽  
Regular Rings

AbstractThe study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules.

scholarly journals RINGS OVER WHICH CYCLICS ARE DIRECT SUMS OF PROJECTIVE AND CS OR NOETHERIAN

2010 ◽  
Vol 52 (A) ◽  
pp. 103-110 ◽  
Author(s):  
C. J. HOLSTON ◽  
S. K. JAIN ◽  
A. LEROY
Keyword(s):  
Projective Module ◽  
Regular Ring ◽  
Artinian Ring ◽  
Von Neumann Regular Ring ◽  
Finitely Generated Module ◽  
Direct Sums ◽  
Von Neumann ◽  
Noetherian Module ◽  
Direct Sum ◽  
Von Neumann Regular

AbstractR is called a right WV-ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right V-ring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. ‘extending modules’) or noetherian module. For a finitely generated module M with projective socle over a V-ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X ⊕ T, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = S ⊕ T, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.

Modules Whose Endomorphism Rings Are (m, n)-Coherent

2019 ◽  
Vol 26 (02) ◽  
pp. 231-242
Author(s):  
Xiaoqiang Luo ◽  
Lixin Mao
Keyword(s):  
Left Ideal ◽  
Endomorphism Ring ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Coherent Ring ◽  
Left Annihilator

Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator [Formula: see text] is a finitely generated left ideal of Mn(S) for any M-m-generated submodule X of Mn if and only if every M-(n, m)-presented right R-module has an add M-preenvelope. As a consequence, we investigate when the endomorphism ring S is left coherent, left pseudo-coherent, left semihereditary or von Neumann regular.

Homological Properties Relative to Injectively Resolving Subcategories

2015 ◽  
Vol 58 (4) ◽  
pp. 741-756 ◽  
Author(s):  
Zenghui Gao
Keyword(s):  
Von Neumann ◽  
Direct Sum ◽  
Von Neumann Regular Rings ◽  
Flat Modules ◽  
Resolving Subcategory ◽  
Von Neumann Regular ◽  
Finitely Presented ◽  
Regular Rings

AbstractLet ε be an injectively resolving subcategory of left R-modules. A left R-module M (resp. right R-module N) is called ε-injective (resp. ε-flat) if Ext1R (G,M) = 0 (resp. TorR1 (N, G) = 0) for any G ∊ ε. Let ε be a covering subcategory. We prove that a left R-module M is E-injective if and only if M is a direct sum of an injective left R-module and a reduced E-injective left R-module. Suppose ℱ is a preenveloping subcategory of right R-modules such that ε+ ⊆ ℱ and ℱ+ ⊆ ε. It is shown that a finitely presented right R-module M is ε-flat if and only if M is a cokernel of an ℱ-preenvelope of a right R-module. In addition, we introduce and investigate the ε-injective and ε-flat dimensions of modules and rings. We also introduce ε-(semi)hereditary rings and ε-von Neumann regular rings and characterize them in terms of ε-injective and ε-flat modules.

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.

scholarly journals On endomorphism rings of Leavitt path algebras

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1175-1181 ◽  
Author(s):  
Tufan Özdin
Keyword(s):  
Endomorphism Ring ◽  
Regular Ring ◽  
The Other ◽  
Endomorphism Rings ◽  
Von Neumann Regular Ring ◽  
Arbitrary Graph ◽  
Von Neumann ◽  
Path Algebras ◽  
Von Neumann Regular ◽  
Leavitt Path Algebras

Let E be an arbitrary graph, K be any field and A be the endomorphism ring of L := LK(E) considered as a right L-module. Among the other results, we prove that: (1) if A is a von Neumann regular ring, then A is dependent if and only if for any two paths in L satisfying some conditions are initial of each other, (2) if A is dependent then LK(E) is morphic, (3) L is morphic and von Neumann regular if and only if L is semisimple and every homogeneous component is artinian.

Modules Whose Endomorphism Rings are Von Neumann Regular

2013 ◽  
Vol 41 (11) ◽  
pp. 4066-4088 ◽  
Author(s):  
Gangyong Lee ◽  
S. Tariq Rizvi ◽  
Cosmin Roman
Keyword(s):  
Endomorphism Rings ◽  
Von Neumann ◽  
Von Neumann Regular

Iso-retractable modules and rings

2019 ◽  
Vol 12 (01) ◽  
pp. 1950013 ◽  
Author(s):  
A. K. Chaturvedi
Keyword(s):  
Endomorphism Ring ◽  
Simple Module ◽  
Sufficient Conditions ◽  
Open Problems ◽  
Von Neumann ◽  
Simple Modules ◽  
Von Neumann Regular

The modules which are isomorphic to their non-zero submodules are known as iso-retractable. We characterize simple modules in terms of iso-retractable modules. We provide several sufficient conditions for iso-retractable modules to be simple. We show that if the endomorphism ring of an iso-retractable module is von-Neumann regular then [Formula: see text] is a simple module. In general, iso-retractable modules need not be projective (injective) and vice versa. We investigate some properties of iso-retractable modules with projectivity as well as injectivity. Finally, we provide some open problems.

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