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 Modules Whose Endomorphism Rings are Baer

2019 ◽  
pp. 1-11
Author(s):  
Thoraya Abdelwhab ◽  
Xiaoyan Yang
Keyword(s):  
Direct Summand ◽  
Endomorphism Rings ◽  
Direct Sum

In this paper, we study modules whose endomorphism rings are Baer, which we call endoBaer modules. We provide some characterizations of endoBaer modules and investigate their properties. Some classes of rings R are characterized in terms of endoBaer R-modules. It is shown that a direct summand of an endoBaer modules inherits the property, while a direct sum of endoBaer modules does not. Necessary and sucient conditions for a nite direct sum of endoBaer modules to be an endoBaer module are provided.

scholarly journals Modules Whose Endomorphism Rings are Right Rickart

2019 ◽  
pp. 1-14
Author(s):  
Thoraya Abdelwhab ◽  
Xiaoyan Yang
Keyword(s):  
Direct Summand ◽  
Endomorphism Ring ◽  
Endomorphism Rings ◽  
Direct Sum

In this paper, we study modules whose endomorphism rings are right Rickart (or right p.p.) rings, which we call R-endoRickart modules. We provide some characterizations of R-endoRickart modules. Some classes of rings are characterized in terms of R-endoRickart modules. We prove that an R-endoRickart module with no innite set of nonzero orthogonal idempotents in its endomorphism ring is precisely an endoBaer module. We show that a direct summand of an R-endoRickart modules inherits the property, while a direct sum of R-endoRickart modules does not. Necessary and sucient conditions for a nite direct sum of R-endoRickart modules to be an R-endoRickart module are provided.

Lifting modules with finite internal exchange property and direct sums of hollow modules

2020 ◽  
pp. 2150189
Author(s):  
Yosuke Kuratomi
Keyword(s):  
Direct Summand ◽  
Sufficient Conditions ◽  
Exchange Property ◽  
Direct Sums ◽  
Perfect Ring ◽  
Direct Sum ◽  
Decomposition Formula ◽  
Direct Sum Decomposition ◽  
Extending Module ◽  
Necessary And Sufficient

A module [Formula: see text] is said to be lifting if, for any submodule [Formula: see text] of [Formula: see text], there exists a decomposition [Formula: see text] such that [Formula: see text] and [Formula: see text] is a small submodule of [Formula: see text]. A lifting module is defined as a dual concept of the extending module. A module [Formula: see text] is said to have the finite internal exchange property if, for any direct summand [Formula: see text] of [Formula: see text] and any finite direct sum decomposition [Formula: see text], there exists a direct summand [Formula: see text] of [Formula: see text] [Formula: see text] such that [Formula: see text]. This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any [Formula: see text]-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.

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 Direct-sum decompositions of modules with semilocal endomorphism rings

2004 ◽  
Vol 274 (2) ◽  
pp. 689-707 ◽  
Author(s):  
Alberto Facchini ◽  
Roger Wiegand
Keyword(s):  
Endomorphism Rings ◽  
Direct Sum

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.

Sign in / Sign up

Export Citation Format

Share Document