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 Relative injectivity and CS-modules

1994 ◽  
Vol 17 (4) ◽  
pp. 661-666
Author(s):  
Mahmoud Ahmed Kamal
Keyword(s):  
Direct Decomposition ◽  
Injective Hull ◽  
Extra Condition ◽  
Direct Sums ◽  
Direct Sum ◽  
Injective Modules ◽  
Semisimple Module ◽  
Continuous Module

In this paper we show that a direct decomposition of modulesM⊕N, withNhomologically independent to the injective hull ofM, is a CS-module if and only ifNis injective relative toMand both ofMandNare CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for quasi-injective modules. But when we confine ourselves to CS-modules we need no conditions on their socles. Then we investigate direct sums of CS-modules which are pairwise relatively inective. We show that every finite direct sum of such modules is a CS-module. This result is known for quasi-continuous modules. For the case of infinite direct sums, one has to add an extra condition. Finally, we briefly discuss modules in which every two direct summands are relatively inective.

scholarly journals Modules Whose Cyclic Submodules Have Finite Dimension

1976 ◽  
Vol 19 (1) ◽  
pp. 1-6 ◽  
Author(s):  
David Berry
Keyword(s):  
Associative Ring ◽  
Finite Dimension ◽  
Essential Extension ◽  
Injective Hull ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Cyclic Submodules ◽  
Unitary Right

R denotes an associative ring with identity. Module means unitary right R-module. A module has finite Goldie dimension over R if it does not contain an infinite direct sum of nonzero submodules [6]. We say R has finite (right) dimension if it has finite dimension as a right R-module. We denote the fact that M has finite dimension by dim (M)<∞.A nonzero submodule N of a module M is large in M if N has nontrivial intersection with nonzero submodules of M [7]. In this case M is called an essential extension of N. N⊆′M will denote N is essential (large) in M. If N has no proper essential extension in M, then N is closed in M. An injective essential extension of M, denoted I(M), is called the injective hull of M.

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.

scholarly journals On τ-completely decomposable modules

2004 ◽  
Vol 70 (1) ◽  
pp. 163-175 ◽  
Author(s):  
Septimiu Crivei
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Torsion Theory ◽  
Positive Answer ◽  
Essential Extension ◽  
Injective Module ◽  
Injective Hull ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Torsion Theories

For a hereditary torsion theory τ, a moduleAis called τ-completedly decomposable if it is a direct sum of modules that are the τ-injective hull of each of their non-zero submodules. We give a positive answer in several cases to the following generalised Matlis' problem: Is every direct summand of a τ-completely decomposable module still τ-completely decomposable? Secondly, for a commutative Noetherian ringRthat is not a domain, we determine those torsion theories with the property that every τ-injective module is an essential extension of a (τ-injective) τ-completely decomposable module.

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.

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.

On essential extensions of direct sums of either injective or projective modules

2014 ◽  
Vol 13 (07) ◽  
pp. 1450038 ◽  
Author(s):  
M. Tamer Koşan ◽  
Truong Cong Quynh
Keyword(s):  
Essential Extension ◽  
Projective Modules ◽  
Direct Sums ◽  
Direct Sum ◽  
Injective Hulls ◽  
Essential Extensions

For a ring R, there are classical facts that R is right Noetherian if and only if every direct sum of injective right R-modules is injective, and R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of injective right R-modules. In this paper, we prove that R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of either injective right R-modules or projective right R-modules.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

scholarly journals Reduction to Dimension Two of the Local Spectrum for an AH Algebra with the Ideal Property

2017 ◽  
Vol 60 (4) ◽  
pp. 791-806 ◽  
Author(s):  
Chunlan Jiang
Keyword(s):  
Inductive Limit ◽  
Local Spectrum ◽  
Direct Sums ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Ideal Property ◽  
Uniformly Bounded ◽  
The Ideal

AbstractA C*-algebra Ahas the ideal property if any ideal I of Ais generated as a closed two-sided ideal by the projections inside the ideal. Suppose that the limit C*-algebra A of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has the ideal property. In this paper we will prove that A can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension-drop interval algebras and matrix algebras over 2-dimensional spaces with torsion H2 groups.

A New Construction of the Injective Hull

1968 ◽  
Vol 11 (1) ◽  
pp. 19-21 ◽  
Author(s):  
Isidore Fleischer
Keyword(s):  
Essential Extension ◽  
Injective Hull ◽  
Minimal Cardinality ◽  
New Construction ◽  
Definition Of

The definition of injectivity, and the proof that every module has an injective extension which is a subextension of every other injective extension, are due to R. Baer [B]. An independent proof using the notion of essential extension was given by Eckmann-Schopf [ES]. Both proofs require the p reliminary construction of some injective overmodule. In [F] I showed how the latter proof could be freed from this requirement by exhibiting a set F in which every essential extension could be embedded. Subsequently J. M. Maranda pointed out that F has minimal cardinality. It follows that F is equipotent with the injective hull. Below Icon struct the injective hull by equipping Fit self with a module strucure.

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