This document summarizes research on the concept of Y-coretractable modules. The main points are:
- An R-module M is Y-coretractable if for every proper y-closed submodule N, there exists a non-zero homomorphism from M/N to M.
- Some basic properties of Y-coretractable modules are investigated, including relationships to other module concepts. Every semisimple, CLS, and singular module is shown to be Y-coretractable.
- The notions of strongly Y-coretractable and mono-Y-coretractable modules are introduced, with characterizations provided. Examples are given to distinguish these concepts.
ABSTRACT--- Let R be a ring with identity and M be an R-module with unitary . In this paper we introduce the
notion of Y-coretractable module . Some basic properties of this class of modules are investigated and some
relationships between these modules and other related concepts are introduced .
Abstract: Let R be a ring with identity and M be an R-module with unity. In this paper we introduce the notion of C-coretractable modules. Some basic properties of this class of modules are investigated and some relationships between these modules and other related concepts are introduced. Also, we give the notion of strongly C-coretractable and study it comparison with C-coretractable.
Keywords and phrases:(strongly) coretractable modules, (strongly) C-coretractable, mono-coretractable, mono-c-coretractable, simple closed, C-Rickart, multiplication module and quasi-Dedekind module.
Abstract: Throughout this paper, all rings have identities and all modules are unitary right modules.
Let R be a ring and M an R-module. A module M is called coretractable if for each proper submodule
N of M, there exists a nonzero homomorphism f from M/N into M. Our concern in this paper is to
develop basic properties of coretractable modules and to look for any relations between coretractable
modules and other classes of modules.
Keyword:. coretractable modules, multiplication modules, quasi-Dedekind modules, rationally–
injective modules, prime module
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
ABSTRACT--- Let R be a ring with identity and M be an R-module with unitary . In this paper we introduce the
notion of Y-coretractable module . Some basic properties of this class of modules are investigated and some
relationships between these modules and other related concepts are introduced .
Abstract: Let R be a ring with identity and M be an R-module with unity. In this paper we introduce the notion of C-coretractable modules. Some basic properties of this class of modules are investigated and some relationships between these modules and other related concepts are introduced. Also, we give the notion of strongly C-coretractable and study it comparison with C-coretractable.
Keywords and phrases:(strongly) coretractable modules, (strongly) C-coretractable, mono-coretractable, mono-c-coretractable, simple closed, C-Rickart, multiplication module and quasi-Dedekind module.
Abstract: Throughout this paper, all rings have identities and all modules are unitary right modules.
Let R be a ring and M an R-module. A module M is called coretractable if for each proper submodule
N of M, there exists a nonzero homomorphism f from M/N into M. Our concern in this paper is to
develop basic properties of coretractable modules and to look for any relations between coretractable
modules and other classes of modules.
Keyword:. coretractable modules, multiplication modules, quasi-Dedekind modules, rationally–
injective modules, prime module
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
Analysis of Cross-ply Laminate composite under UD load based on CLPT by Ansys...IJERA Editor
In current study the strength of composite material configuration is obtained from the properties of constituent
laminate by using classical laminate plate theory. For the purpose of analysis various configurations of 2 layered
and 4 layered cross ply laminates are used. The material of laminate is supposed to be boron/epoxy having
orthotropic properties. The loading in current study is supposed to be of uniformly distributed load type. For the
analysis purpose software working on finite element analysis logics i.e. Ansys mechanical APDL is used. By the
help of Ansys mechanical APDL the deflection and stress intensity is found out. The effect of variation of
laminate layers is also studied in current study along with the effect of variation of stacking patterns. The current
study will also help to conclude which stacking pattern is best in 2 layered and 4 layered cross ply laminate.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
In this paper , some results about the concept of completely coretractable ring are studied and also another results about
the coretractable module are recalled and investigated.
Some Remarks on Prime Submodules in Weak Co-Multiplication ModulesIJERA Editor
Throughout this paper, the entire ring will be treated as commutative ring with non-zero identity and all the
modules will be treated as unitary modules. In this paper we shall discuss some remarks on prime submodules in
weak co-multiplication modules.
Analysis of Cross-ply Laminate composite under UD load based on CLPT by Ansys...IJERA Editor
In current study the strength of composite material configuration is obtained from the properties of constituent
laminate by using classical laminate plate theory. For the purpose of analysis various configurations of 2 layered
and 4 layered cross ply laminates are used. The material of laminate is supposed to be boron/epoxy having
orthotropic properties. The loading in current study is supposed to be of uniformly distributed load type. For the
analysis purpose software working on finite element analysis logics i.e. Ansys mechanical APDL is used. By the
help of Ansys mechanical APDL the deflection and stress intensity is found out. The effect of variation of
laminate layers is also studied in current study along with the effect of variation of stacking patterns. The current
study will also help to conclude which stacking pattern is best in 2 layered and 4 layered cross ply laminate.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
In this paper , some results about the concept of completely coretractable ring are studied and also another results about
the coretractable module are recalled and investigated.
Some Remarks on Prime Submodules in Weak Co-Multiplication ModulesIJERA Editor
Throughout this paper, the entire ring will be treated as commutative ring with non-zero identity and all the
modules will be treated as unitary modules. In this paper we shall discuss some remarks on prime submodules in
weak co-multiplication modules.
Some Remarks on Prime Submodules in Weak Co-Multiplication ModulesIJERA Editor
Throughout this paper, the entire ring will be treated as commutative ring with non-zero identity and all the
modules will be treated as unitary modules. In this paper we shall discuss some remarks on prime submodules in
weak co-multiplication modules.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Relative superior mandelbrot and julia sets for integer and non integer valueseSAT Journals
Abstract
The fractals generated from the self-squared function,
2 zz c where z and c are complex quantities have been studied
extensively in the literature. This paper studies the transformation of the function , 2 n zz c n and analyzed the z plane and
c plane fractal images generated from the iteration of these functions using Ishikawa iteration for integer and non-integer values.
Also, we explored the drastic changes that occurred in the visual characteristics of the images from n = integer value to n = non
integer value.
Keywords: Complex dynamics,
Relative Superior Julia set, Relative Superior Mandelbrot set.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Comparative study of results obtained by analysis of structures using ANSYS, ...IOSR Journals
The analysis of complex structures like frames, trusses and beams is carried out using the Finite
Element Method (FEM) in software products like ANSYS and STAAD. The aim of this paper is to compare the
deformation results of simple and complex structures obtained using these products. The same structures are
also analyzed by a MATLAB program to provide a common reference for comparison. STAAD is used by civil
engineers to analyze structures like beams and columns while ANSYS is generally used by mechanical engineers
for structural analysis of machines, automobile roll cage, etc. Since both products employ the same fundamental
principle of FEM, there should be no difference in their results. Results however, prove contradictory to this for
complex structures. Since FEM is an approximate method, accuracy of the solutions cannot be a basis for their
comparison and hence, none of the varying results can be termed as better or worse. Their comparison may,
however, point to conservative results, significant digits and magnitude of difference so as to enable the analyst
to select the software best suited for the particular application of his or her structure.
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented as block diagonal matrices (resulting in the Wedderburn decomposition), general forms of polyadic structures are given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures (“double” decomposition of two kinds). We then introduce the polyadization concept (a “polyadic constructor”), according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The “deformation” by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
In silico drugs analogue design: novobiocin analogues.pptx
4691 15817-1-pb
1. Asian Journal of Applied Sciences (ISSN: 2321 – 0893)
Volume 05 – Issue 02, April 2017
724)www.ajouronline.comAsian Online Journals (
Y-coretractable and Strongly Y-coretractable Modules
Inaam Mohammed Ali Hadi1
, Shukur Neamah Al-Aeashi2,*
1
Department Of Mathematics, College 0f Education for Pure Sciences (Ibn-Al-Haitham)
University Of Baghdad , Iraq
2
Department Of Urban Planning , College 0f Physical Planning
University Of Kufa , Iraq
*
Corresponding author’s email: shukur.mobred [AT] uokufa.edu.iq
____________________________________________________________________________________________________________
ABSTRACT--- Let R be a ring with identity and M be an R-module with unitary . In this paper we introduce the
notion of Y-coretractable module . Some basic properties of this class of modules are investigated and some
relationships between these modules and other related concepts are introduced .
Keywords--- coretractable module , Y-coretractable module , C-coretractable module
______________________________________________________________________________________________
1. INTRODUCTION
Throughout this paper all rings have identities and all modules are unital right R-modules . A module M is called
coretractable if for each a proper submodule N of M , there exists a nonzero R-homomorphism f:M/N→M [1] , and an R-
module M is called strongly coretractable module if for each proper submodule N of M , there exists a nonzero R-
homomorphism f:M/N→M such that Imf+N=M [2]. It is clear every strongly coretractable module is coretractable but it
is not conversely [2]. In this paper, we introduce the notion of Y-coretractable module where an R-module M is called Y-
coretractable if for each proper y-closed submodule N of M , there exists a nonzero homomorphism f HomR(M/N,M) . It
is clear every coretractable module is Y-coretractable but it is not conversely. In this work we study some basic
properties of this notion and give the notion of strongly Y-coretractable and mono-Y-coretractable modules . A
characterizations of these modules are given . Some examples and remarks are given and some important
characterizations and properties are investigated . Also we study its relation with other concepts . We recall the following
concepts a submodule N of M is called y-closed submodule if M/N is nonsingular module . And every y-closed
submodule is closed but not conversely, for example < > in Z6 is closed , not y-closed submodule [8,P.19] . If a module
is nonsingular , then the concepts closed and y-closed are coincide [3] . Note that asgari and Haghany in[4], introduced
the concept of t-closed submodule . However the two concepts t-closed and y-closed submodule are coincide [see
Proposition(2.6)] in [5]. An R-module M is called CLS-module if every y-closed submodule is direct summand
submodule [12] . Also asgari and Haghany in [5] , introduced the concept of t-extending , where a module is t-extending
if every t-closed submodule is direct summand . Thus t-extending and CLS-module are coincide too . However M is a
nonsingular R-module . Then M is extending module if and only if M is CLS-module [12] . End(M) is the set of
endomorphisms of M .
2. Y- CORETRACTABLE MODULES
Definition(1.1): An R-module is called Y-coretractable module if for each y-closed proper submodule N of M , there
exists a nonzero homomorphism f Hom(M/N, M) .
Equivalently , M is an Y-coretractable R-module if for each proper y-closed submodule N of M , there exists
g EndR(M) , g 0 and g(N)=0 . A ring R is called Y-coretractable if R is Y-coretractable R-module .
Examples and Remarks(1.2)
(1) It is clear that every coretractable module is Y-coretractable . But the converse is not true , for example Z as Z-
module is Y-coretractable , but it is not coretractable .
(2) Y-coretractability is not preserved by taking submodules , factor modules and direct summands , since for any
R-module M and a cogenerator R-module C , C M is a cogenerator by [1], and so C M is a coretractable module .
2. Asian Journal of Applied Sciences (ISSN: 2321 – 0893)
Volume 05 – Issue 02, April 2017
724)www.ajouronline.comAsian Online Journals (
Thus C M is Y-coretractable , but M need not be Y-coretractable . Where an R-module M is called cogenerator if for
every nonzero homomorphism f:M1→ M2 where M1 and M2 are R-modules , g:M2→M such that g◦f ≠ 0 .
(3) Every C-coretractable R-module is Y-coretractable module . Where an R-module M is called C-coretractable
module if for each proper closed submodule N of M , there exists a nonzero homomorphism f HomR(M/N,M) [3].
Proof : It is clear , since every y-closed submodule is closed .
(4) Every semisimple module is Y-coretractable . For example Z2 Z2, Z6 , … are semisimple module and hence Y-
coretractable , but the converse is not true for example the Z-module Z2 Z8 is C-coretractable , and so Y-coretractable ,
but it is not semisimple .
(5) Every CLS-module is an Y-coretractable module .
Proof : Since M is CLS-module , so every y-closed submodule is a direct summand and hence it is clear that M is Y-
coretractable module .
For example , the Z-module M=Q Z2 is CLS-module , hence M is Y-coretractable module .
(6) Every singular R-module is Y-coretractable module .
Proof: Since every singular module is CLS-module by [8,Remark(2.3.3]. Hence by Examples and Remarks(1.2(5)) , then
the result is hold ◙
For example M= Z4 Z2 as Z-module is singular , hence it is Y-coretractable .
(7) Let M M' where M is an Y-coretractable R-module . Then M' is Y-coretractable .
Proof : Since M M' , so there exists f:M M' be R-isomorphism . Let W be a proper y-closed submodule of M' . Then
N=f-1
(W) is proper y-closed submodule of M . Since M is Y-coretractable module . So there exists a nonzero R-
homomorphism h:M/N M such that h(M/N)+N=M . Now, the mapping f◦g◦f-1
EndR(M'). Moreover, f◦g◦f-1
(M')=f◦g(f-
1
(M'))= f◦g(M)=f(g(M)) . But g(M) 0 and f is an isomorphism , so f(g(M)) 0 . Thus f◦g◦f-1
(M') 0. Also f◦g◦f-1
(W) =
f◦g(f-1
(W)) = f◦g(N)= f(g(N)) = f(0) =0 . Thus M' is Y-coretractable .
(8) Every extending module M is an Y-coretractable .
Proof : Let M be an extending module . Thus M is a C-coretractable by [3] and hence it is Y-coretractable
◙
We can prove the following two results .
Proposition(1.3): A fully invariant direct summand N of Y-coretractable R-module M is a Y-coretractable module .
Proof : Since N is a direct summand of M , so there exists a submodule W of M such that N W=M . Let K be a proper
closed submodule of N , then K W is a closed submodule in N W=M . Since M is a Y-coretractable module , so
there exists f EndR(M) , f≠0 and f(K W) =0 . Suppose that g is a restriction map from N into M , g≠0 . Also N is fully
invariant direct submodule . Then N is stable module [5,Lemma(2.1.6)] . So g(N) N . Therefore g EndR(N) , g≠0 .
g(K)=f|N(K)=0. Thus N is a Y-coretractable ◙
Corollary(1.4): Let N be a direct summand submodule of duo R-module M . If M is a Y-coretractable . Then N is also
Y-coretractable .
Recall that an R-module M is called a generalized extending module if for any submodule N of M , there exists a
direct summand K of M such that N K and K/N is singular [6] , and An R-module M is called Y-extending if for every
submodule N of M , there exists a direct summand D of M such that N D≤eN and N D≤eD [7] .
Proposition(1.5): Let M be an R-module . If M satisfies any one of the following conditions :
(1) M is a generalized extending module ;
(2) M is an Y-extending module .
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Then M is an Y-coretractable module
Proof : If condition(1) hold . Since every generalized extending module is CLS-module by [8,Proposition(2.3.13)]) .
Thus M is Y- coretractable . If condition(2) hold . Since every Y-extending module is CLS-module by [8,
Proposition(2.3.14) , P.44] . Hence M is Y-coretractable , by Examples and Remarks (1.2(5))
◙
Proposition(1.6): Let M be a nonsingular R-module .Then M is Y-coretractable module if and only if M is C-
coretractable module .
Proof : It is clear , since the two concepts closed and y-closed submodules are coincide in the class of nonsingular
modules ◙
Proposition(1.7): Let M , A and B be R-modules . Consider the short exact 0 → A B M→0 . If f(A) is an essential
submodule of B , then M is Y-coretractable module .
Proof : Since f(A) is an essential submodule of B . Hence B/f(A) is singular module by [9, page32] . But g is an
epimorphism which implies B/kerg M . But exactness of the sequence kerg=f(A). Hence B/f(A) M . Therefore M is
singular module , and so by Examples and Remarks (1.2(6)) , M is Y-coretractable module
◙
Corollary(1.8): Let M be an R-module . Then M/N is Y-coretractable module for each proper essential submodule N of
M .
Proof : Suppose that N be a proper essential submodule of M . Consider the short exact 0→N M M/N→0 where π is
the natural epimorphism from M into M/N . Then by Proposition(1.7) , we get M/N is Y-coretractable module .
We get the following immediately by Corollary(1.8)
Corollary(1.9): Let M be an R-module. If M is a uniform module . Then M/N is an Y-coretractable module for each
submodule N of M .
As a consequence of Corollary(1.8) and Corollary(1.9) , the Z-module Q/Z is Y-coretractable .
Proposition(1.10): Let M be a projective R-module . If A E(M) such that A M is y-closed submodule of M .Then M
is an Y-coretractable module .
Proof : The hypothesis implies M is a CLS-module by [8,Proposition(2.4.3)] . Hence by Examples and Remarks(1.2(5)),
M is an Y-coretractable module
Proposition(1.11): Let M be an R-module . If M is Y-coretractable and for any proper submodule N of M , there exists
y-closed submodule W of M such that N W . Then M is a coretractable .
Proof : Let N be a proper submodule . By hypothesis there exists y-closed submodule W of M such that N W . But M is
Y-coretractable , hence there exists f EndR(M) , f≠0 and f(W)=0 which implies that f(N)=0 . Therefore M is
coretractable module
We can get easily the following two results by similar to [10,Theorem(1.11)] and [10,Theorem(1.13)] , respectively .
Theorem(1.12): Let M= , where Mα is an R-module for all α I such that every y-closed submodule of M is
fully invariant submodule. If Mα is Y-coretractable module for all α I , then M is a Y-coretractable .
Theorem(1.13): Let M1 and M2 be R-modules , M=M1 M2 such that ann(M1)+ann(M2)=R . If M1 and M2 are Y-
coretractable module , then M is a Y-coretractable module .
Proposition(1.14): Every quasi-Dedekind Y-coretractable R-module is nonsingular .
Proof : Let N be a proper y-closed submodule of M . Then M/N is nonsingular . As M is Y-coretractable , there exists f
EndR(M) , f≠0 and N kerf . But kerf=0 ( since M is quasi-Dedekind ) , so N=0 . It follows that M/(0) M , so M is
nonsingular module .
3. MONO-Y-CORETRACTABLE MODULES
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Before we introduce this notion we need to show two concepts , an R-module M is called mono-C-coretractable if for
each proper closed submodule of M , there exists f EndR(M) , f≠0 and N=kerf [4] and M is called mono-coretractable if
for each proper submodule of M , there exists f EndR(M) , f≠0 and N=kerf . Clearly that every mono-coretractable
module is mono-C-coretractable but not conversely [4] .
Definition(2.1): An R-module M is called mono-Y-coretractable if for each proper y-closed submodule N of M , there
exists f EndR(M) , f≠0 and N=kerf .
Examples and Remarks(2.2):
(1) Every mono-coretractable is mono-Y-coretractable but not conversely , for example , Z as Z-module is mono-
Y-coretractable , but not mono-coretractable .
(2) Every mono-C-coretractable module implies mono-Y-coretractable.
(3) Every semisimple module is mono-Y-coretractable .
Definition(2.3): An R-module M is called Y-fully stable if for each y-closed submodule is stable .
It is clear every fully stable is Y-fully stable , but not conversely . For example , M=Z8 Z2 as Z-module . The only y-
closed submodule is M and M is stable , so M is Y-fuuly stable . But f:N→M , where N=<( , )>= { ( , ) , ( , ),
( , ), ( , ) } , define by f((( , ))= ( , ) for all ( , ) N . It is clear that f(N) N , so M is not fully stable
Proposition(2.4): Let M be a strongly Rickart R-module . Then M is mono-Y-coretractable if and only if M is Y-fully
stable CLS-module .
Proof : ( ) Let N be a proper y-closed submodule of M . Since M is mono-Y-coretractable module , so there exists f
EndR(M) , f≠0 and kerf=N . But M is strongly Rickart , so kerf is stable direct summand . Thus M is Y-fully stable and M
is CLS-module .
( ) It is clear .
Now , we can prove the following Propositions by similar proof of [10,Proposition(3.6)] .
Proposition(2.5): Let M be a mono-Y-coretractable Y-fully stable R-module . Then every nonzero y-closed submodule
of M is also mono-Y-coretractable module .
Proposition(2.6): Let M=M1 M2 where M1 and M2 be R-modules such that every y-closed submodule of M is fully
invariant . If M1 and M2 are mono-Y-coretractable modules , then M is mono-Y-coretractable .
Proof : Let K be a proper y-closed submodule of M . Since K is fully invariant by hypothesis, K= K1 K2 where K1 is y-
closed submodule of M1 and K2 is y-closed submodule of M2 .
Case(1): K1 is a proper submodule of M1 and K2 = M2 . Since M1 is mono-Y-coretractable module , so there exists
f EndR(M1) f 0 and kerf= K1 . Define g:M M by g((x,y))=(f(x),0) for each (x,y) M . It is clear that g EndR(M) ,
g 0 and kerg=K .
Case(2): If K1=M1 and K2 is a proper submodule of M2, then by a similar proof of case(1) , we can get the result .
Case(3): K1 is a proper submodule of M1 and K2 is a proper submodule of M2 . Since M1 and M2 are mono-Y-
coretractable module , then there exist f EndR(M) , g EndR(M) such that f 0 and g 0 . Define h EndR(M) by h((x,y))
= (f(x),g(y)) for each (x,y) M . Then h EndR(M) , h 0 and kerh=K .
Now , we shall present another generalization for the concept strongly cortractable as the following :
4. STRONGLY Y- CORETRACTABLE MODULES
Definition(3.1): An R-module M is called strongly Y-coretractable module if for each proper y-closed submodule N of
M , there exists a nonzero homomorphism f HomR(M/N,M) and Imf +N =M .
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Equivalently , M is strongly Y-coretractable R-module if for each proper y-closed submodule N of M , there exists
g EndR(M) , g 0 , g(N)=0 and Imf+N=M . A ring R is called strongly Y-coretractable if R is strongly Y-coretractable
R-module .
Examples and Remarks(3.2):
(1) Every semisimple module is a strongly Y-coretractable .
(2) Every CLS-module is a strongly Y-coretractable . For example , Z4 as Z4-module has no proper y-closed , so Z4
is CLS-module and hence strongly Y-coretractable module .
(3) Every singular R-module is a strongly Y-coretractable .
(4) Every strongly coretractable is a strongly Y-coretractable . But the converse is not true in general . For example
consider M=Z12 as Z-module is not strongly coretractable module by part(1) , but M is singular module , and hence M is
a strongly Y-coretractable by part(3).
(5) Every strongly C-coretractable R-module is strongly Y-coretractable.
Proof : It is clear since every y-closed submodule is closed . ◙
(6) If M be a faithful multiplication module over CLS-ring, then M is a strongly Y-coretractable module .
Proof : By [8, Proposition(2.3.12),P.43] , M is CLS-module . Hence by part (2) , M is strongly Y-coretractable .
(7) Let M be a nonsingular R-module . Then M is strongly Y-coretractable module if and only if M is strongly C-
coretractable .
Proof : It is clear since y-closed and closed submodule are coinside under nonsingular module .
Proposition(3.3): Let M M' where M is a strongly Y-coretractable R-module . Then M' is strongly Y-coretractable
module
Proof : Since M M' , so there exists f:M M' be R-isomorphism . Let W be a proper y-closed submodule of M' . Then
N=f-1
(W) is a proper y-closed submodule of M . Since M is strongly Y-coretractable module . So there exists a nonzero
R-homomorphism h:M/N M such that h(M/N)+N=M . And we get easily that M' is a strongly Y-coretractable R-
module ◙
Proposition(3.4): Let M be a strongly Y-coretractable R-module and N be a proper Y-closed submodule of M ,then M/N
is strongly Y-coretractable module .
Proof : Let W/N be a proper y-closed submodule of M/N . Since N is y-closed submodule of M , so W is y-closed
submodule of M. But M is strongly Y-coretractable module , hence there exists a nonzero R-homomorphism g:M/W M
and g(M/W)+W=M . But
(M/N)/(W/N) M/W . Set f= ᵒg where is the natural epimorphism from M into M/W . Then f( ) + = (g( )) +
= + = = = .
we can get M/N is also strongly Y-coretractable module . ◙
Proposition(3.5): If M is CLS-module , then M/N is a strongly Y- coretractable module for each N≤M .
Proof : Let N be a proper submodule of M . Since M is CLS-module , hence M/N is CLS-module by [8,
Proposition(2.3.11) ,P.42] and hence M/N is strongly Y-coretractable module by Examples and Remarks(3.2 (2))
◙
Recall that , let M be an R-module . The second singular submodule Z2(M) is a submodule in M containing Z(M)
such that Z2(M)/Z(M) is singular submodule of M/ Z(M) [8] . Consider Z-module Q. Z2(Q)=0 . Consider Z-module Zn .
Z2(Zn)=Zn .
Proposition(3.6): Let M be an R-module , then M/Z2(M) is a strongly Y- coretractable module .
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Proof: Since Z2(M) is y-closed submodule of M by [8,Remark(2.2.3), P.32] , then M/Z2(M) is CLS-module by [8,
Proposition(2.3.11) , P.42] , and hence it is strongly Y-coretractable module by Examples and Remarks(3.2 (2)) .
Proposition(3.7): Let M=M1 M2 , where M is duo module (or distributive or annM1+annM2 = R) . Then M is strongly
Y-coretractable module , if M1 and M2 are strongly Y-coretractable module .
Proof : Let N be a proper y-closed submodule of M . Since M is duo or distributive , then N=(N M1) (N M2) . Thus
N=N1 N2 for some N1≤M1 and N2 ≤ M2 . Thus each of N1 and N2 are y-closed submodules M1 and M2 respectively.
Case(1): K1 is a proper submodule of M1 and K2 is a proper submodule of M2 . Since M1 and M2 are strongly
coretractable module , then there exists f : M1/K1 M1 such that Imf+K1=M1 , and there exists g:M2/K2 M2 such that
Img+K2=M2 .
Now, define a nonzero R-homomorphism h:M/K→M ;
that is h:(M1 M2)/(K1 K2) M1 M2 by h[(m1,m2)+K1 K2]=[f(m1 K1) , g(m2 K2)] . Then h is well-defined . Now
, Imh+K=Imh +(K1 K2)=(Imf Img)+ (K1 K2) =(Imf+K1) (Img+K2) =M1 M2=M .
Case(2): K1=M1 and K2 is a nonzero proper submodule of M2 .
Consider (M1 M2)/(K1 K2)=(M1 M2)/(M1 K2) M2/K2 . Since M2 is strongly coretractable module , so there exists
f:M2/K2 M2 such that Imf + K2 = M2 . Define g : M2/ K2 M1 M2 by g = i◦f where i is the inclusion mapping from
M2 into M1 M2 . Therefore Img+K= Img+ (K1 K2) = Img+(M1 K2) = M1 (Img+K2) = M1 ((0) Imf)+K2) =
M1 ((0)+(Imf+K2) = M1 ((0)+M2) =M1 M2 = M .
Case(3): K1 is a nonzero proper submodule of M1 and K2 = M2 , then by a similar proof case(2) , we can get the result
Case(4): K1=0 and K2 = M2 .
Consider M/K (M1 M2)/(0 K2) M1 . Let i is the inclusion mapping from M1 into M1 M2 , hence i(M1)+K=( M1
0) (0 M2)=M
Case(5): K1=M1 and K2=0 , then the proof is similar to Case(4) .
M is a strongly Y-coretractable module . ◙
Now, we can get the following result directly by Proposition(1.5) and Remark and Example(3.2 (2)) :
Proposition(3.8): Let M be an R-module . Then M is strongly Y-coretractable module if M satisfies any one of the
following conditions :
(1) M is generalized extending module ;
(2) M is Y-extending module ;
Also , we can get the following result by a similar proof of Proposition(1.7) :
Proposition(3.9): Let M,A and B be R-modules with the short exact 0 → A B M→0 . If f(A) is an essential
submodule of B , then M is strongly Y-coretractable module .
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