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?

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 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.

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.

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.

The Torsion Submodule of A Cyclic Module Splits Off

1972 ◽  
Vol 24 (3) ◽  
pp. 450-464 ◽  
Author(s):  
Mark L. Teply
Keyword(s):  
Direct Summand ◽  
Integral Domain ◽  
Torsion Theory ◽  
Cyclic Module ◽  
Torsion Module ◽  
Associative Rings

A prominent question in the study of modules over an integral domain has been: “When is the torsion submodule t(A) of a module A a direct summand of A?” A module is said to split when its torsion module is a direct summand. Clearly, every cyclic module over an integral domain splits. Interesting splitting problems have been explored by Kaplansky [14; 15], Rotman [20], Chase [4], and others.Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Two of the most important of these concepts are Goldie's torsion theory (see [1; 12; 22]) and the simple torsion theory (see [5; 6; 8; 9; 23], and their references).

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.

Divisibility of Direct Sums in Torsion Theories

1976 ◽  
Vol 28 (1) ◽  
pp. 211-214 ◽  
Author(s):  
B. Sarath ◽  
K. Varadarajan
Keyword(s):  
Sufficient Conditions ◽  
Torsion Theory ◽  
Necessary And Sufficient Conditions ◽  
Countable Subset ◽  
Direct Sums ◽  
Direct Sum ◽  
Hereditary Torsion Theory ◽  
Torsion Theories ◽  
Necessary And Sufficient

Given a hereditary torsion theory on the category Mod R of right R-modules we obtain in this paper necessary and sufficient conditions for the direct sum of a given family of R-modules to be divisible for the torsion theory . Using this criterion we show that if is a family of R-modules having the property that is divisible for every countable subset K ol J then is itself divisible.

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.

scholarly journals Hereditary Torsion Theory Counter Equivalences

1996 ◽  
Vol 183 (1) ◽  
pp. 217-230 ◽  
Author(s):  
R.R. Colby ◽  
K.R. Fuller
Keyword(s):  
Torsion Theory ◽  
Hereditary Torsion Theory

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.

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