On H-Supplemented Modules

2011 ◽  
Vol 18 (spec01) ◽  
pp. 915-924 ◽  
Author(s):  
Derya Keskin Tütüncü ◽  
Mohammad Javad Nematollahi ◽  
Yahya Talebi
Keyword(s):  
Direct Summand ◽  
Torsion Theory ◽  
Direct Sum

Let [Formula: see text]be a finite direct sum of modules. We prove: (i) If Mi is radical Mj-projective for all j > i and each Mi is H-supplemented, then M is H-supplemented. (ii) If all the Mi are relatively projective and N is H-supplemented, then each Mi is H-supplemented. Let ρ be the preradical for a cohereditary torsion theory. Let M be a module such that ρ (M) has a unique coclosure and every direct summand of ρ (M) has a coclosure in M. Then M is H-supplemented if and only if there exists a decomposition M=M1⊕M2 such that M2 ⊆ ρ(M), ρ(M)/M2 ≪ M/M2, and M1, M2 are H-supplemented.

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.

DECOMPOSITIONS OF MODULES SUPPLEMENTED RELATIVE TO A TORSION THEORY

2005 ◽  
Vol 16 (01) ◽  
pp. 43-52
Author(s):  
M. TAMER KOŞAN ◽  
ABDULLAH HARMANCI
Keyword(s):  
Direct Summand ◽  
Torsion Theory ◽  
Torsion Module ◽  
Direct Sum ◽  
Hereditary Torsion Theory ◽  
Semisimple Module ◽  
Torsion Functor

Let R be a ring, M a right R-module and a hereditary torsion theory in Mod-R with associated torsion functor τ for the ring R. Then M is called τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. In [4], M is called almost τ-torsion if every proper submodule of M is τ-torsion. We present here some properties of these classes of modules and look for answers to the following questions posed by the referee of the paper [4]: (1) Let a module M = M′ ⊕ M″ be a direct sum of a semisimple module M′ and τ-supplemented module M″. Is M τ-supplemented? (2) Can one find a non-stable hereditary torsion theory τ and τ-supplemented modules M′ and M″ such that M′ ⊕ M″ is not τ-supplemented? (3) Can one find a stable hereditary torsion theory τ and a τ-supplemented module M such that M/N is not τ-supplemented for some submodule N of M? (4) Let τ be a non-stable hereditary torsion theory and the module M be a finite direct sum of almost τ-torsion submodules. Is M τ-supplemented? (5) Do you know an example of a torsion theory τ and a τ-supplemented module M with τ-torsion submodule τ(M) such that M/τ(M) is not semisimple?

scholarly journals On a class of dual Rickart modules

2020 ◽  
Vol 72 (7) ◽  
pp. 960-970
Author(s):  
R. Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Finite Set ◽  
Rickart Modules

UDC 512.5 Let R be a ring and let Ω R be the set of maximal right ideals of R . An R -module M is called an sd-Rickart module if for every nonzero endomorphism f of M , ℑ f is a fully invariant direct summand of M . We obtain a characterization for an arbitrary direct sum of sd-Rickart modules to be sd-Rickart. We also obtain a decomposition of an sd-Rickart R -module M , provided R is a commutative noetherian ring and A s s ( M ) ∩ Ω R is a finite set. In addition, we introduce and study ageneralization of sd-Rickart modules.

t-Rickart and Dual t-Rickart Modules

2015 ◽  
Vol 22 (spec01) ◽  
pp. 849-870 ◽  
Author(s):  
Sh. Asgari ◽  
A. Haghany
Keyword(s):  
Direct Summand ◽  
Torsion Module ◽  
Direct Sum ◽  
Rickart Modules ◽  
Baer Modules ◽  
Equivalent Conditions

We introduce the notion of t-Rickart modules as a generalization of t-Baer modules. Dual t-Rickart modules are also defined. Both of these are generalizations of continuous modules. Every direct summand of a t-Rickart (resp., dual t-Rickart) module inherits the property. Some equivalent conditions to being t-Rickart (resp., dual t-Rickart) are given. In particular, we show that a module M is t-Rickart (resp., dual t-Rickart) if and only if M is a direct sum of a Z2-torsion module and a nonsingular Rickart (resp., dual Rickart) module. It is proved that for a ring R, every R-module is dual t-Rickart if and only if R is right t-semisimple, while every R-module is t-Rickart if and only if R is right Σ-t-extending. Other types of rings are characterized by certain classes of t-Rickart (resp., dual t-Rickart) modules.

scholarly journals Rings characterised by semiprimitive modules

1995 ◽  
Vol 52 (1) ◽  
pp. 107-116
Author(s):  
Yasuyuki Hirano ◽  
Dinh Van Huynh ◽  
Jae Keol Park
Keyword(s):  
Direct Summand ◽  
Projective Module ◽  
Injective Module ◽  
Direct Sum

A module M is called a CS-module if every submodule of M is essential in a direct summand of M. It is shown that a ring R is semilocal if and only if every semiprimitive right R-module is CS. Furthermore, it is also shown that the following statements are equivalent for a ring R: (i) R is semiprimary and every right (or left) R-module is injective; (ii) every countably generated semiprimitive right R-module is a direct sum of a projective module and an injective module.

scholarly journals Decompositions of modules over a discrete valuation ring

1979 ◽  
Vol 27 (3) ◽  
pp. 284-288 ◽  
Author(s):  
Robert O. Stanton
Keyword(s):  
Direct Summand ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Torsion Module ◽  
Direct Sum ◽  
Rank One ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractLet N be a direct summand of a module which is a direct sum of modules of torsion-free rank one over a discrete valuation ring. Then there is a torsion module T such that N⊕T is also a direct sum of modules of torsion-free rank one.

On Purifiable Subsocles of a Primary Abelian Group

1971 ◽  
Vol 23 (1) ◽  
pp. 48-57 ◽  
Author(s):  
John Irwin ◽  
James Swanek
Keyword(s):  
Abelian Group ◽  
Direct Summand ◽  
Pure Subgroup ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Interesting Connection

In this paper we shall investigate an interesting connection between the structure of G/S and G, where S is a purifiable subsocle of G. The results are interesting in the light of a counterexample by Dieudonné [3, p. 142] who exhibits a primary abelian group G, where G/S is a direct sum of cyclic groups, but G is not a direct sum of cyclic groups. Surprisingly, the assumption of the purifiability of S allows G to inherit the structure of G/S. In particular, we show that if G/S is a direct sum of cyclic groups and S supports a pure subgroup H, then G is a direct sum of cyclic groups and if is a direct summand of G which is of course a direct sum of cyclic groups. It is also shown that if G/S is a direct sum of torsion-complete groups and S supports a pure subgroup H, then G is a direct sum of torsion-complete groups and H is a direct summand of G, and is also a direct sum of torsion-complete groups.

scholarly journals On indecomposable decompositions of CS-modules

1996 ◽  
Vol 61 (1) ◽  
pp. 30-41 ◽  
Author(s):  
Nguyen V. Dung
Keyword(s):  
Direct Summand ◽  
Infinite Sequence ◽  
Exchange Property ◽  
Endomorphism Rings ◽  
Direct Sum

AbstractIt is shown that, over any ring R, the direct sum M = ⊕i∈IMi of uniform right R-modules Mi with local endomorphism rings is a CS-module if and only if every uniform submodule of M is essential in a direct summand of M and there does not exist an infinite sequence of non-isomorphic monomorphisms , with distinct in ∈ I. As a consequence, any CS-module which is a direct sum of submodules with local endomorphism rings has the exchange property.

scholarly journals Classes of modules with many direct summands

1995 ◽  
Vol 59 (1) ◽  
pp. 8-19 ◽  
Author(s):  
I. Al-Khazzi ◽  
P. F. Smith
Keyword(s):  
Direct Summand ◽  
Krull Dimension ◽  
Direct Sum ◽  
Semisimple Module

AbstractLet R be any ring with identity, M a unital right R-module and α ≥ 0 an ordinal. Then M is a direct sum of a semisimple module and a module having Krull dimension at most α if and only if for every submodule N of M there exists a direct summand K of M such that K ⊆ N and N/K has Krull dimension at most α.

On coseparable and 𝔪-coseparable modules

2020 ◽  
pp. 2150031
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Rings ◽  
Direct Products ◽  
Finitely Generated ◽  
Direct Sums ◽  
Direct Sum ◽  
Free Modules

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

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