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.

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.

Σ-Rickart modules

2019 ◽  
Vol 19 (11) ◽  
pp. 2050207
Author(s):  
Gangyong Lee ◽  
Mauricio Medina-Bárcenas
Keyword(s):  
Direct Summand ◽  
Endomorphism Ring ◽  
Finitely Generated ◽  
Hereditary Ring ◽  
Direct Sum ◽  
Factor Module ◽  
Rickart Modules

Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50’s. In this paper, we study the notion of a [Formula: see text]-Rickart module by utilizing the endomorphism ring of a module and using the recent notion of a Rickart module, as a module theoretic analogue of a right hereditary ring. A module [Formula: see text] is called [Formula: see text]-Rickart if every direct sum of copies of [Formula: see text] is Rickart. It is shown that any direct summand and any direct sum of copies of a [Formula: see text]-Rickart module are [Formula: see text]-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of hereditary rings: a ring [Formula: see text] is right hereditary if and only if every submodule of any projective right [Formula: see text]-module is projective if and only if every factor module of any injective right [Formula: see text]-module is injective. Also, we have a characterization of a finitely generated [Formula: see text]-Rickart module in terms of its endomorphism ring. Examples which delineate the concepts and results are provided.

scholarly journals KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

2010 ◽  
Vol 52 (A) ◽  
pp. 69-82 ◽  
Author(s):  
ALBERTO FACCHINI ◽  
ŞULE ECEVIT ◽  
M. TAMER KOŞAN
Keyword(s):  
Local Ring ◽  
Endomorphism Ring ◽  
Weak Form ◽  
Indecomposable Module ◽  
Endomorphism Rings ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Injective Modules ◽  
Maximal Ideals

AbstractWe show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull–Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum ofnkernels of morphisms between injective indecomposable modules can have exactlyn! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. IfERis an injective indecomposable module andSis its endomorphism ring, the duality Hom(−,ER) transforms kernels of morphismsER→ERinto cyclically presented left modules over the local ringS, sending the monogeny class into the epigeny class and the upper part into the lower part.

Endoprime modules and their direct sums

2018 ◽  
Vol 17 (08) ◽  
pp. 1850155 ◽  
Author(s):  
Gangyong Lee ◽  
S. Tariq Rizvi
Keyword(s):  
Direct Summand ◽  
Prime Ring ◽  
Endomorphism Ring ◽  
Matrix Ring ◽  
Direct Sums ◽  
Direct Sum ◽  
Finite Matrix ◽  
Reduced Ring ◽  
Baer Modules ◽  
Special Classes

The purpose of this paper is to further study the endoprime modules as one of the special classes of quasi-Baer modules. As a module theoretic analogue of a prime ring, we characterize an endoprime module via its endomorphism ring and a weak retractability condition. It is shown that any direct summand of an endoprime module is an endoprime module. A characterization is obtained when a direct sum of endoprime modules is an endoprime module. It is well known that every prime ring is semicentral reduced. We prove that a column (and row) finite matrix ring over a semicentral reduced ring is also a semicentral reduced ring. Consequently, it is shown that a column (and row) finite matrix ring over a prime ring is prime. Applications and examples illustrating our results are provided.

scholarly journals Finite torsion-free rank endomorphism rings

2015 ◽  
Vol 31 (1) ◽  
pp. 39-43
Author(s):  
SIMION BREAZ ◽  
Keyword(s):  
Abelian Group ◽  
Small Group ◽  
Endomorphism Ring ◽  
Endomorphism Rings ◽  
Direct Sum ◽  
Bounded Group ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

We prove that a finite torsion-free rank abelian group with finite torsion-free rank endomorphism ring is a direct sum of a bounded group and a self-small group.

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 The characterization problem for endomorphism rings

1991 ◽  
Vol 50 (1) ◽  
pp. 116-137 ◽  
Author(s):  
J. L. García
Keyword(s):  
Endomorphism Ring ◽  
Free Module ◽  
Endomorphism Rings ◽  
Characterization Problem ◽  
Algebraic Properties

AbstractWe consider the problem of characterizing by abstract properties the rings which are isomorphic to the endomorphism ring End (RF) of some free module F over a ring R in a given class R of rings. We solve this problem when R is any class of rings (by employing topological notions) and when R is the class of all the left Kasch rings (in terms of algebraic properties only).

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.

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