On Hermitian endomorphism rings

2017 ◽  
Vol 16 (11) ◽  
pp. 1750220
Author(s):  
Wanru Zhang
Keyword(s):  
Endomorphism Ring ◽  
Necessary Conditions ◽  
Exchange Property ◽  
Projective Modules ◽  
Endomorphism Rings ◽  
Dual Problems ◽  
Injective Modules ◽  
Sufficient And Necessary Conditions

Let [Formula: see text] be a right [Formula: see text]-module with finite exchange property and let [Formula: see text] be its endomorphism ring. In this paper, some sufficient and necessary conditions for [Formula: see text] to be a Hermitian ring are given. Moreover, we investigate Hermitian endomorphism rings of quasi-projective modules by means of completions of diagrams. The dual problems for quasi-injective modules are also studied.

Endomorphism Rings and Gabriel Topologies

1984 ◽  
Vol 36 (2) ◽  
pp. 193-205 ◽  
Author(s):  
Soumaya Makdissi Khuri
Keyword(s):  
Endomorphism Ring ◽  
Lattice Isomorphism ◽  
Close Connection ◽  
Projective Modules ◽  
Finitely Generated ◽  
Basic Tool ◽  
Endomorphism Rings ◽  
Finite Dimensional ◽  
Faithful Module ◽  
Correspondence Theorem

A basic tool in the usual presentation of the Morita theorems is the correspondence theorem for projective modules. Let RM be a left R-module and B = HomR(M, M). When M is a progenerator, there is a close connection (in fact a lattice isomorphism) between left R-submodules of M and left ideals of B, which can be applied to the solution of problems such as characterizing when the endomorphism ring of a finitely generated projective faithful module is simple or right Noetherian. More generally, Faith proved that this connection can be retained in suitably modified form when M is just a generator in R-mod ([4], [2], [3]). In this form the correspondence theorem can be applied to show, e.g., that, when RM is a generator, then (a): RM is finite-dimensional if and only if B is a left finite-dimensional ring and in this case d(RM) = d(BB), and (b): If RM is nonsingular then B is a left nonsingular ring ([6]).

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.

scholarly journals QF-3 endomorphism rings of Σ-quasi-projective modules

1988 ◽  
Vol 30 (2) ◽  
pp. 215-220 ◽  
Author(s):  
José L. Gómez Pardo ◽  
Nieves Rodríguez González
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Sufficient Conditions ◽  
Torsion Theory ◽  
Projective Modules ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Quotient Module ◽  
Qf Ring ◽  
Necessary And Sufficient

A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End(RM) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].

scholarly journals Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1323
Author(s):  
Shyam Sundar Santra ◽  
Rami Ahmad El-Nabulsi ◽  
Khaled Mohamed Khedher
Keyword(s):  
Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Mixed Delays ◽  
Neutral Differential Equations ◽  
Canonical Operator ◽  
Second Order Differential Equations ◽  
The Future ◽  
Sufficient And Necessary Conditions

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

scholarly journals Properties of Certain Subclass of Meromorphic Multivalent Functions Associated with q-Difference Operator

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1035
Author(s):  
Cai-Mei Yan ◽  
Rekha Srivastava ◽  
Jin-Lin Liu
Keyword(s):  
Difference Operator ◽  
Necessary Conditions ◽  
Partial Sums ◽  
Coefficient Estimates ◽  
Radius Of Starlikeness ◽  
Closure Theorems ◽  
Sufficient And Necessary Conditions ◽  
Distortion Theorems ◽  
Growth And Distortion Theorems

A new subclass Σp,q(α,A,B) of meromorphic multivalent functions is defined by means of a q-difference operator. Some properties of the functions in this new subclass, such as sufficient and necessary conditions, coefficient estimates, growth and distortion theorems, radius of starlikeness and convexity, partial sums and closure theorems, are investigated.

Perfect colorings of patterns with multiple orbits

2019 ◽  
Vol 75 (6) ◽  
pp. 814-826
Author(s):  
Allan Junio ◽  
Ma. Lailani Walo
Keyword(s):  
Necessary Conditions ◽  
Sufficient And Necessary Conditions

This paper studies colorings of patterns with multiple orbits, particularly those colorings where the orbits share colors. The main problem is determining when such colorings become perfect. This problem is attacked by characterizing all perfect colorings of patterns through the construction of sufficient and necessary conditions for a coloring to be perfect. These results are then applied on symmetrical objects to construct both perfect and non-perfect colorings.

scholarly journals On a Periodic Predator-Prey System with Holling III Functional Response and Stage Structure for Prey

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Xiangzeng Kong ◽  
Zhiqin Chen ◽  
Li Xu ◽  
Wensheng Yang
Keyword(s):  
Functional Response ◽  
Prey Species ◽  
Necessary Conditions ◽  
Stage Structure ◽  
Predator Prey ◽  
Prey System ◽  
Sufficient And Necessary Conditions ◽  
Holling Iii Functional Response

We propose and study the permanence of the following periodic Holling III predator-prey system with stage structure for prey and both two predators which consume immature prey. Sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.

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