On Direct Sums of Injective Modules and Chain Conditions

1994 ◽  
Vol 46 (3) ◽  
pp. 634-647 ◽  
Author(s):  
Stanley S. Page ◽  
Yiqiang Zhou
Keyword(s):  
Full Subcategory ◽  
Natural Class ◽  
Direct Sums ◽  
Chain Conditions ◽  
Injective Modules ◽  
Injective Hulls ◽  
Equivalent Conditions

AbstractLet R be a ring and M a right R-module. Let σ[M] be the full subcategory of Mod-R subgenerated by M. An M-natural class 𝒦 is a subclass of σ[M] closed under submodules, direct sums, isomorphic copies, and M-injective hulls. We present some equivalent conditions each of which describes when σ has the property that direct sums of (M-)injective modules in σ are (M-)injective. Specializing to particular M, and/or special subclasses we obtain many new results and known results as corollaries.

scholarly journals Relative continuity of direct sums of M-injective modules

2000 ◽  
Vol 62 (1) ◽  
pp. 51-56
Author(s):  
Liu Zhongkui ◽  
Javed Ahsan
Keyword(s):  
Natural Class ◽  
Direct Sums ◽  
Direct Sum ◽  
Injective Modules

Let M be a left R-module and  be an M-natural class with some additional conditions. It is proved that every direct sum of M-injective left R-modules in  is  -continuous (-quasi-continuous) if and only if every direct sum of M- injective left R-modules in  is M-injective.

scholarly journals Relative natural classes and relative injectivity

2005 ◽  
Vol 2005 (5) ◽  
pp. 671-678
Author(s):  
Septimiu Crivei
Keyword(s):  
Torsion Theory ◽  
Natural Class ◽  
Direct Sums ◽  
Hereditary Torsion Theory ◽  
Injective Hulls ◽  
Natural Classes

Letτbe a hereditary torsion theory on the categoryR-Mod of leftR-modules over an associative unitary ringR. We introduce the notion ofτ-natural class as a class of modules closed underτ-dense submodules, direct sums, andτ-injective hulls. We study connections between certain conditions involvingτ-(quasi-)injectivity in the context ofτ-natural classes, generalizing results established by S. S. Page and Y. Q. Zhou (1994) for natural classes.

scholarly journals INJECTIVE MODULES OVER DOWN-UP ALGEBRAS

2010 ◽  
Vol 52 (A) ◽  
pp. 53-59 ◽  
Author(s):  
PAULA A. A. B. CARVALHO ◽  
CHRISTIAN LOMP ◽  
DILEK PUSAT-YILMAZ
Keyword(s):  
Roots Of Unity ◽  
Finiteness Conditions ◽  
Simple Modules ◽  
Injective Modules ◽  
Injective Hulls

AbstractThe purpose of this paper is to study finiteness conditions on injective hulls of simple modules over Noetherian down-up algebras. We will show that the Noetherian down-up algebras A(α, β, γ) which are fully bounded are precisely those which are module-finite over a central subalgebra. We show that injective hulls of simple A(α, β, γ)-modules are locally Artinian provided the roots of X2 − αX − β are distinct roots of unity or both equal to 1.

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 RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES

2012 ◽  
Vol 54 (3) ◽  
pp. 605-617 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
NOYAN ER ◽  
NİL ORHAN ERTAŞ
Keyword(s):  
Prime Ideals ◽  
Cyclic Module ◽  
Direct Sums ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Injective Modules ◽  
Dedekind Domains ◽  
Singular Ring

AbstractDedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

SOME PROPERTIES ON ISOCLINISM OF LIE ALGEBRAS AND COVERS

2008 ◽  
Vol 07 (04) ◽  
pp. 507-516 ◽  
Author(s):  
ALI REZA SALEMKAR ◽  
HADI BIGDELY ◽  
VAHID ALAMIAN
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Schur Multiplier ◽  
Direct Sums ◽  
Schur Multipliers ◽  
Finite Dimensional ◽  
Equivalent Conditions

In this paper, we give some equivalent conditions for Lie algebras to be isoclinic. In particular, it is shown that if two Lie algebras L and K are isoclinic then L can be constructed from K and vice versa using the operations of forming direct sums, taking subalgebras, and factoring Lie algebras. We also study connection between isoclinic and the Schur multiplier of Lie algebras. In addition, we deal with some properties of covers of Lie algebras whose Schur multipliers are finite dimensional and prove that all covers of any abelian Lie algebra have Hopfian property. Finally, we indicate that if a Lie algebra L belongs to some certain classes of Lie algebras then so does its cover.

Strongly Gorenstein categories

2021 ◽  
pp. 2250213
Author(s):  
Wan Wu ◽  
Zenghui Gao
Keyword(s):  
Direct Summand ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Abelian Category ◽  
Direct Products ◽  
Direct Sums ◽  
Gorenstein Categories

We introduce and study strongly Gorenstein subcategory [Formula: see text], relative to an additive full subcategory [Formula: see text] of an abelian category [Formula: see text]. When [Formula: see text] is self-orthogonal, we give some sufficient conditions under which the property of an object in [Formula: see text] can be inherited by its subobjects and quotient objects. Then, we introduce the notions of one-sided (strongly) Gorenstein subcategories. Under the assumption that [Formula: see text] is closed under countable direct sums (respectively, direct products), we prove that an object is in right (respectively, left) Gorenstein category [Formula: see text] (respectively, [Formula: see text]) if and only if it is a direct summand of an object in right (respectively, left) strongly Gorenstein subcategory [Formula: see text] (respectively, [Formula: see text]). As applications, some known results are obtained as corollaries.

An equivalence criterion for the generalized injectivity of modules with respect to algebraic classes of homomorphisms

2016 ◽  
Vol 15 (09) ◽  
pp. 1650166 ◽  
Author(s):  
Jorge E. Macías-Díaz ◽  
Siegfried Macías
Keyword(s):  
General Definition ◽  
Injective Modules ◽  
Equivalence Criterion ◽  
Definition Of ◽  
Injective Hulls ◽  
Essential Extensions ◽  
The Way

Departing from a general definition of injectivity of modules with respect to suitable algebraic classes of morphisms, we establish conditions under which two modules are isomorphic when they are isomorphic to submodules of each other. The main result of this work extends both Bumby’s criterion for the isomorphism of injective modules and the well-known Cantor–Bernstein–Schröder’s theorem on the cardinality of sets. In the way, various properties on essential extensions, injective modules and injective hulls are generalized. The applicability of our main theorem embraces the cases of [Formula: see text]-injective and pure-injective modules as particular scenarios. Many of the propositions which lead to the proof of the main result of this paper are valid for arbitrary categories.

Sign in / Sign up

Export Citation Format

Share Document