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.
Fixed Point Theorem in Fuzzy Metric SpaceIJERA Editor
In this present paper on fixed point theorems in fuzzy metric space . we extended to Fuzzy Metric space
generalisation of main theorem .
Mathematics Subject Classification: 47H10, 54A40
PROBABILISTIC INTERPRETATION OF COMPLEX FUZZY SETIJCSEIT Journal
The innovative concept of Complex Fuzzy Set is introduced. The objective of the this paper to investigate
the concept of Complex Fuzzy set in constraint to a traditional Fuzzy set , where the membership function
ranges from [0, 1], but in the Complex fuzzy set extended to a unit circle in a complex plane, where the
member ship function in the form of complex number. The Compressive study of mathematical operation of
Complex Fuzzy set is presented. The basic operation like addition, subtraction, multiplication and division
are described here. The Novel idea of this paper to measure the similarity between two fuzzy relations by
evaluating δ -equality. Here also we introduce the probabilistic interpretation of the complex fuzzy set
where we attempted to clarify the distinction between Fuzzy logic and probability.
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and computer graphics and circular models, related to angular / periodic data, have vide applications in various walks of life. Visualizing a circular model through THPD matrix the required computations on THPD matrices using single bordering and double bordering are discussed. It can be seen that every tridiagonal THPD leads to Cardioid model.
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning. Part 1 covers: 1. A brief history of Random Matrix Theory, 2. Classical Random Matrix Ensembles (basic building blocks)
IOSR Journal of Electronics and Communication Engineering(IOSR-JECE) is an open access international journal that provides rapid publication (within a month) of articles in all areas of electronics and communication engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in electronics and communication engineering. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
On The Transition Probabilities for the Fuzzy States of a Fuzzy Markov ChainIJERA Editor
In this paper the theory of fuzzy logic is mixed with the theory of Markov systems and the abstraction of a
Markov system with fuzzy states introduced. The notions such as fuzzy transient, fuzzy recurrent etc., were
introduced. The results based on these notions are introduced.
Design of Digital Adder Using Reversible LogicIJERA Editor
Reversible logic circuits have promising applications in Quantum computing, Low power VLSI design,
Nanotechnology, optical computing, DNA computing and Quantum dot cellular automata. In spite of them
another main prominent application of reversible logic is Quantum computers where the quantum devices are
essential which are ideally operated at ultra high speed with less power dissipation must be built from reversible
logic components. This makes the reversible logic as a one of the most promising research areas in the past few
decades. In VLSI design the delay is the one of the major issue along with area and power. This paper presents
the implementation of Ripple Carry Adder (RCA) circuits using reversible logic gates are discussed.
An Efficient Photo Voltaic System for Onboard Ship ApplicationsIJERA Editor
In this paper a high efficient photovoltaic system is proposed for onboard ship applications which convert the
lower voltage obtained from solar modules to higher voltage required by the ship service loads. In a typical
photovoltaic system only step-up /boost converter is used due to which the converter has to operate in extreme
duty ratio, resulting in increase of switching losses and thus decreasing the overall efficiency. But in this paper
the conventional boost converter is used with interleaved inductors and capacitors. The poposed system stores
the energy in inductors and thus reduces the stress in the switches (Without allowing the total voltage to appear
across the switch). The simulation is designed using MATLAB/Simulink with an Input voltage of 40-V to
achieve a output voltage of 300-380 V. The developed simulation results are compared for output powers of
500W and 1kW
Study on Theoretical Aspects of Virtual Data Integration and its ApplicationsIJERA Editor
Data integration is the technique of merging data residing at different sources at different locations, and
providing users with an integrated, reconciled view of these data. Such unified view is called global or mediated
schema. It represents the intentional level of the integrated and reconciled data. In the data integration system,
our area of interest in this paper is characterized by an architecture based on a global schema and a set of
sources or source schemas. The objective of this paper is to provide a study on the theoretical aspects of data
integration systems and to present a comprehensive review of the applications of data integration in various
fields including biomedicine, environment, and social networks. It also discusses a privacy framework for
protecting user’s privacy with privacy views and privacy policies.
Groundwater Quality Assessment in hard rock terrain of Rasipuram Taluk, Namak...IJERA Editor
Groundwater is of most important to rural development in many countries of the world. Over exploitation of
groundwater has become a major challenge not only to the present civilization and also for the future
generations. The main focus of this study is to assess the suitability of groundwater quality for drinking and
irrigation purposes in vicinity of Rasipuram block in Tamil Nadu. Groundwater samples from 15 locations were
collected from different wells during January 2015 and analyzed for different physico-chemical parameters. The
usefulness of these parameters in predicting groundwater quality characteristics were discussed. The quality of
groundwater in the study area is fresh to brackish water, moderately hard to very hard in nature. The piper plot
shows that the most of the groundwater samples fall in the field of Na+-Cl- and mixed Ca++-Na+-Cl- type. Water
quality index rating was carried out to quantify overall groundwater quality status of the area. The WQI for these
samples ranges from 37.34 to 650. Hence majority of the water samples are poor to very poor in water quality.
The area in general is characterized by hard water, hence is not suitable for drinking purpose. The samples
plotted in the piper and USSL diagram were used to understand the chemical characteristic of groundwater for
irrigation purposes. However, the values of SAR, Na% and RSC indicate that groundwater is suitable for
irrigation purposes. Overall water quality of the study area was found satisfactory for drinking purpose except in
few locations and suitable for irrigation purpose. Hence the local government needs to initiate remedial
measures.
Fixed Point Theorem in Fuzzy Metric SpaceIJERA Editor
In this present paper on fixed point theorems in fuzzy metric space . we extended to Fuzzy Metric space
generalisation of main theorem .
Mathematics Subject Classification: 47H10, 54A40
PROBABILISTIC INTERPRETATION OF COMPLEX FUZZY SETIJCSEIT Journal
The innovative concept of Complex Fuzzy Set is introduced. The objective of the this paper to investigate
the concept of Complex Fuzzy set in constraint to a traditional Fuzzy set , where the membership function
ranges from [0, 1], but in the Complex fuzzy set extended to a unit circle in a complex plane, where the
member ship function in the form of complex number. The Compressive study of mathematical operation of
Complex Fuzzy set is presented. The basic operation like addition, subtraction, multiplication and division
are described here. The Novel idea of this paper to measure the similarity between two fuzzy relations by
evaluating δ -equality. Here also we introduce the probabilistic interpretation of the complex fuzzy set
where we attempted to clarify the distinction between Fuzzy logic and probability.
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and computer graphics and circular models, related to angular / periodic data, have vide applications in various walks of life. Visualizing a circular model through THPD matrix the required computations on THPD matrices using single bordering and double bordering are discussed. It can be seen that every tridiagonal THPD leads to Cardioid model.
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning. Part 1 covers: 1. A brief history of Random Matrix Theory, 2. Classical Random Matrix Ensembles (basic building blocks)
IOSR Journal of Electronics and Communication Engineering(IOSR-JECE) is an open access international journal that provides rapid publication (within a month) of articles in all areas of electronics and communication engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in electronics and communication engineering. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
On The Transition Probabilities for the Fuzzy States of a Fuzzy Markov ChainIJERA Editor
In this paper the theory of fuzzy logic is mixed with the theory of Markov systems and the abstraction of a
Markov system with fuzzy states introduced. The notions such as fuzzy transient, fuzzy recurrent etc., were
introduced. The results based on these notions are introduced.
Design of Digital Adder Using Reversible LogicIJERA Editor
Reversible logic circuits have promising applications in Quantum computing, Low power VLSI design,
Nanotechnology, optical computing, DNA computing and Quantum dot cellular automata. In spite of them
another main prominent application of reversible logic is Quantum computers where the quantum devices are
essential which are ideally operated at ultra high speed with less power dissipation must be built from reversible
logic components. This makes the reversible logic as a one of the most promising research areas in the past few
decades. In VLSI design the delay is the one of the major issue along with area and power. This paper presents
the implementation of Ripple Carry Adder (RCA) circuits using reversible logic gates are discussed.
An Efficient Photo Voltaic System for Onboard Ship ApplicationsIJERA Editor
In this paper a high efficient photovoltaic system is proposed for onboard ship applications which convert the
lower voltage obtained from solar modules to higher voltage required by the ship service loads. In a typical
photovoltaic system only step-up /boost converter is used due to which the converter has to operate in extreme
duty ratio, resulting in increase of switching losses and thus decreasing the overall efficiency. But in this paper
the conventional boost converter is used with interleaved inductors and capacitors. The poposed system stores
the energy in inductors and thus reduces the stress in the switches (Without allowing the total voltage to appear
across the switch). The simulation is designed using MATLAB/Simulink with an Input voltage of 40-V to
achieve a output voltage of 300-380 V. The developed simulation results are compared for output powers of
500W and 1kW
Study on Theoretical Aspects of Virtual Data Integration and its ApplicationsIJERA Editor
Data integration is the technique of merging data residing at different sources at different locations, and
providing users with an integrated, reconciled view of these data. Such unified view is called global or mediated
schema. It represents the intentional level of the integrated and reconciled data. In the data integration system,
our area of interest in this paper is characterized by an architecture based on a global schema and a set of
sources or source schemas. The objective of this paper is to provide a study on the theoretical aspects of data
integration systems and to present a comprehensive review of the applications of data integration in various
fields including biomedicine, environment, and social networks. It also discusses a privacy framework for
protecting user’s privacy with privacy views and privacy policies.
Groundwater Quality Assessment in hard rock terrain of Rasipuram Taluk, Namak...IJERA Editor
Groundwater is of most important to rural development in many countries of the world. Over exploitation of
groundwater has become a major challenge not only to the present civilization and also for the future
generations. The main focus of this study is to assess the suitability of groundwater quality for drinking and
irrigation purposes in vicinity of Rasipuram block in Tamil Nadu. Groundwater samples from 15 locations were
collected from different wells during January 2015 and analyzed for different physico-chemical parameters. The
usefulness of these parameters in predicting groundwater quality characteristics were discussed. The quality of
groundwater in the study area is fresh to brackish water, moderately hard to very hard in nature. The piper plot
shows that the most of the groundwater samples fall in the field of Na+-Cl- and mixed Ca++-Na+-Cl- type. Water
quality index rating was carried out to quantify overall groundwater quality status of the area. The WQI for these
samples ranges from 37.34 to 650. Hence majority of the water samples are poor to very poor in water quality.
The area in general is characterized by hard water, hence is not suitable for drinking purpose. The samples
plotted in the piper and USSL diagram were used to understand the chemical characteristic of groundwater for
irrigation purposes. However, the values of SAR, Na% and RSC indicate that groundwater is suitable for
irrigation purposes. Overall water quality of the study area was found satisfactory for drinking purpose except in
few locations and suitable for irrigation purpose. Hence the local government needs to initiate remedial
measures.
Experimental Study on Two-Phase Flow in Horizontal Rectangular Minichannel wi...IJERA Editor
An experimental study was conducted to investigate two-phase air-water flow characteristics, in horizontal
rectangular minichannel with Y-junction. The width (W), the height (H) and the hydraulic diameter (DH) of the
rectangular cross section for the upstream side of the junction are 4.60 mm, 2.50 mm and 3.24 mm, while those
for the downstream side are 2.36 mm, 2.50 mm and 2.43 mm. The entire test section was machined from
transparent acrylic block, so that the flow structure could be visualized. Liquid single-phase and air-liquid twophase
flow experiments were conducted at room temperature. The flow pattern, the bubble velocity, the bubble
length, and the void fraction were measured with a high-speed video camera. Pressure profile upstream and
downstream from the junction was also measured for the respective flows, and the pressure loss due to the
contraction at the junction was determined from the pressure profiles. Two flow patterns, i.e., slug and annular
flows, were observed in the fully-developed region apart from the junction. In the analysis, the frictional pressure
drop data, the two-phase frictional multiplier data, bubble velocity data, bubble length data and void fraction data
were compared with calculations by some correlations in literatures. In addition, new pressure loss coefficient
correlations for the pressure drop at the junction has been proposed. Results of such experiment and analysis are
described in the present paper.
Effect of Algal Bio-fertilizer on the Vigna radiata: A Critical ReviewIJERA Editor
The continuous increasing demand of food crops and decrease in productivity due to continuous use of chemical
fertilizer has not only resulted in decline of crop yield, loss of fertility and degradation of soil but has also led us
one step back in achieving sustainable agriculture. The use of algal bio-fertilizer provides an effective, ecofriendly
and non-polluting approach in improving the productivity of crop by both nitrogen fixation and
photosynthesis. Algal bio-fertilizers improve soil structure and increase yield productivity even if applied in a
small area. The application of algal bio-fertilizers in plants has resulted in increase in root, shoot length with
number of leaves and hence overall growth of the plant has been increased. India being one of the largest
producer and consumer of pulses requires abundant amount of pulse production to fulfil the demands of ever
growing populations which can be achieved by using algal bio-fertilizers. This paper briefly underlines the usage
of algal bio-fertilizers as an important tool for sustainability and alternative usage against the chemical
fertilizers.
Nanofluid Flow past an Unsteady Permeable Shrinking Sheet with Heat Source or...IJERA Editor
The consideration of nanofluids has been paid a good attention on the forced convection; the analysis focusing
nanofluids in porous media are limited in literature. Thus, the use of nanofluids in porous media would be very
much helpful in heat and mass transfer enhancement. In this paper, the influence of variable suction, Newtonian
heating and heat source or sink heat and mass transfer over a permeable shrinking sheet embedded in a porous
medium filled with a nanofluid is discussed in detail. The solutions of the nonlinear equations governing the
velocɨty, temperature and concentration profiles are solved numerically using Runge-Kutta Gill procedure
together with shooting method and graphical results for the resulting parameters are displayed and discussed.
The influence of the physical parameters on skin-friction coefficient, local Nusselt number and local Sherwood
number are shown in a tabulated form.
اجکل جو اسلام کو مغربی خواہشات کے مطابق بدلنا چاہتا ہے وہ جھٹ سے کہتا ہے "ہمیں اجتہاد کرنا چاہئے" ان سب کی رہنمائی کیلئے اجتہاد کی آسان انداز میں تفصیل۔ یہ درست ہے کہ امر کو اجتہاد کی ضرورت ہے لیکن اسلئے نہیں کہ جہاد، حدود،کو ختم کیا جائے، یا سود کو جائز کہا جائے یا خاندانی منصوبہ بندی کو حلال کیا جائے۔ جی اسلئے نہیں۔
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.
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
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.
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.
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler totient function, polyadic division with a remainder, etc. are defined. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the binary limit, and the corresponding finite polyadic rings are introduced. Further, polyadic versions of (prime) finite fields are considered. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that any finite polyadic field should contain a certain canonical prime polyadic field as a smallest finite subfield, which can be considered a polyadic analogue of GF (p).
On essential pseudo principally-injective modulestheijes
Pseudo-injectivity is a generalization of quasi- injectivity. An essential Pseudo-injective module wasintroduced by R. Jaiswal, P.C. Bharadwaj and S. Wongwai [7]. This paper is generalization of above notion with new properties
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.
NEW NON-COPRIME CONJUGATE-PAIR BINARY TO RNS MULTI-MODULI FOR RESIDUE NUMBER ...csandit
In this paper a new Binary-to-RNS converters for multi-moduli RNS based on conjugate-pair as
of the set { 2
n1
– 2, 2
n1
+ 2, 2
n2
– 2, 2
n2
+ 2, …, 2
nN
– 2, 2
nN
+ 2 } are presented. 2
n
– 2
and 2
n
+ 2 modulies are called conjugates of each other. Benefits of Multi-moduli RNS
processors are; relying on the sets with pairs of conjugate moduli : 1) Large dynamic ranges. 2)
Fast and balanced RNS arithmetic. 3) Simple and efficient RNS processing hardware. 4)
Efficient weighted-to-RNS and RNS-to-Weighted converters. [1] The dynamic range (M)
achieved by the set above is defined by the least common multiple (LCM) of the moduli. This
new non-coprime conjugate-pair is unique and the only one of its shape as to be shown.
Prime Ring With 3n1 d Contained In The Nucleus.IJRES Journal
In this paper we show that if R is a non associative ring with a derivation d then
d R N n ( ) 3 1
and ( ) , 3 1 Rd R N n
using this it is show that if R is a non associative prime ring such
that d (R) N. n Where n is a fixed positive integer then either R is associative or thederivatives which are
in arithmetic progression becomes zero i.e 3n1 d =0.
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.
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About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Some Remarks on Prime Submodules in Weak Co-Multiplication Modules
1. Arvind Kumar Sinha Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 2, (Part - 1) February 2016, pp.110-114
www.ijera.com 110 | P a g e
Some Remarks on Prime Submodules in Weak Co-Multiplication
Modules
Arvind Kumar Sinha*
, Dibyendu Sasmal**
*
Assistant Professor, Department of Mathematics, National Institute of Technology Raipur, Chhattisgarh,
INDIA.
**
Project Fellow, Department of Mathematics, National Institute of Technology Raipur, Chhattisgarh, INDIA.
ABSTRACT
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.
Mathematical Subject classification: 13C05, 13C13, 13A15, 13C99, 13E05, 16D10, 16D70.
Keywords - Multiplication Modules, Co-multiplication Modules, Weak Multiplication Modules, Weak co-
multiplication Modules, Prime Submodules.
I. INTRODUCTION
In the theory of modules, prime submodules of
modules are very important concept in an algebraic
structure and have obtained a very important
attention in the society of research of this field.
In 1981, Barnad [26] has given the concept of
multiplication modules. After using the concept
prime submodules of modules, the concept of weak
multiplication module has been given by Azizi Shiraz
[6] in the year 2003. Ansari-Toroghy and Farshadifer
[8] has introduced the notion co-multiplication
module as a dual notion of multiplication module in
2007. Using the concept of prime submodule of
modules the concept of weak co-multiplication
module has been given by Atani and Atani [7] in the
year 2009. So in that way we can say that the notion
co-multiplication module is the dual of the notion
weak multiplication module. The aim of this paper is
to provide more information in this field and to
discuss some results on prime submodules in weak
co-multiplication modules.
II. PRELIMINARIES
Throughout this paper, we consider all rings as
commutative rings with identities and all modules as
unital modules. The notation Z will denote the ring of
integers and Q will denote the ring of rational
numbers. In this section, we give some basic
definitions and results for further understanding.
If N and K are submodules of R-module M then
the residual ideal N by K is defined as an ideal
(N :R K) = {r ϵ R | rK N}.
In a special case in which N= 0, the ideal
(0 :R K) is called annihilator of K and it is denoted
by Ann R (K). Also the submodule (0 :M I) is called
the annihilator of I in M and it is denoted by Ann n
(I).
If R is a ring and N is a submodule of an
R-module M, the ideal {r ϵ R | rM N} will be
denoted by [N : M]. Then [0 : M] is the annihilator
of M, Ann (M).
Definition 2.1 [1]: An R-module M is called a
multiplication module if for each submodule N of
M, N = IM, for some ideal I of R. In this case we
can take I = [N : M].
It is easy to see that in this case N = [N: M] M,
where [N : M] = Ann (M/N). (see [17])
Clearly, M is a multiplication module if and only if
for each mM, Rm = [Rm : M] M. (see [2])
For an R-module M, we define the ideal
(M) = Mm
M]:[Rm .
If M is multiplication then M = Mm
Rm
= Mm
M]M:[Rm
= ( Mm
M]:[Rm ) M
= (M) M.
Moreover, if N is a submodule of M, then
N = [N : M] M
= [N : M] (M)M
= (M) [N : M] M
= (M) N. (See [3])
Example 2.2 [4]: Examples of multiplication ideals
(i.e. ideals of a rings R that are multiplication R-
modules) include invertible ideals, principal ideals
and ideals generated by idempotent.
RESEARCH ARTICLE OPEN ACCESS
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Sang Cheol Lee, Sunah Kim and Sang-Cho
Chung [5] has defined multiplication module in the
term of extended as follows:
Definition 2.3 [5]: Let R be a commutative ring and
let M be an R-module. Then a submodule N of M is
said to be extended if N = IM, for same ideal I of R.
Definition 2.4 [5]: M is called multiplication module
if every submodule of M is extended. For example,
every proper submodule of the Z-module Z(p∞
) is a
multiplication module but the Z-module Z(p∞
) itself
is not a multiplication module.
In H. Ansari - Toroghy and F. Farshadifar [8]
the notion of a co-multiplication module was
introduced as a dual of the concepts of a
multiplication module.
Definition 2.5 [5]: An R-module m is defined to be a
co-multiplication module if for each submodule N
of M, N = (0 :M I) for same ideal I of R.
In this care we can take I = Ann (N).
Example 2.6: The Z-module Z2
∞
is co-
multiplication module since all of its proper
submodules are of the form (0 :M 2k
Z) for K =
0,1,2,3,4,……
It is clear that M is co multiplication module if and
only if for every submodule N of M, we have Ann M
(Ann R (N)) = N. (see[9])
Example 2.7: Z4 is a co-multiplication Z-module.
Consider the Z-module M = Z / 4Z and set N = 2Z /
4Z. Then N and M/N are co-multiplication
Z-modules. (see [20])
An R-module M is called co-multiplication
module if for any submodule N of M, there exists an
ideal I of such that N = Ann M (I). (see [11] [12])
An R-module M is co-multiplication module if and
only if for any submodule N of M, N = (0 :M Ann
(N)) (see [8] )
Every proper submodule of a co-multiplication
module is co-multiplication module. (see [8])
Example 2.8: If V is a two dimensional vector space
over a filed K then V cannot be co-multiplication
module but every proper submodule as everyone-
dimensional vector space is co-multiplication
module. (see [11])
Example 2.9 [8]: Let p be any prime number. Let
M = Z (p∞
). Then M is a co-multiplication Z-module
where Z is the ring of integers.
Proof: Fix a prime integer p. Define a set
Qp = { r / pt
| r, t Z}
Then, Qp is additive abelian group containing Z.
Define a set M = Z (p∞
) = Qp / Z.
Then, M is a Z- module. Let N be any submodule of
M. Then N = Z(1/pi
+ Z) , for some integer i.( i ≥ 0 ).
Set I = pi
Z.
Now, Ann M (pi
Z) = N. Hence M is a co-
multiplication Z-module.
Now here is an example which shows that not
every R-module is co-multiplication module.
Example 2.10 [11]: Consider Z as a Z-module. Now,
2Z is a submodule of Z, we have
Ann Z (2Z) = {mZ | 2Zm = 0} = (0).
Now, if Z is a co-multiplication module then by [8],
we have
2Z = ( 0 : Z AnnZ (2Z))
But (0 :Z AnnZ (2Z)) = (0 :Z 0 ) = Z ≠ 2Z
Therefore, Z is not a co-multiplication module.
As in [8] the notion of co-multiplication module
was introduced as a dual of the concept of a
multiplication module. An R-module M is called co-
multiplication (co-m for short) if for every
submodule N of M, there exists an Ideal I of R such
that N = (0 :M I).
It is clear that M is co-m if and only if for every
submodule N of M, we have Ann M (Ann R (N)) = N.
Definition 2.11 [10]: A proper submodule N of M is
said to be prime submodule if from r m N for r
R and m M, We can deduce that r (N : M) or
m N. (see for example in [10])
Definition 2.12 [6]: An R-module M is called weak
multiplication module if M doesn’t have any prime
submodule or every prime submodule N of M We
have N = IM where I is an ideal of R.
Example 2.13 [6]: Q is a weak multiplication
Z-module, which is not a multiplication Z-module.
Every finitely generated weak multiplication module
is a multiplication module. (see [6]).
In 2009 Reza Ebrahimi Atani and Shahabaddin
Ebrahimi Atani [7] have given the notion weak co-
multiplication modules as the dual notion of weak
multiplication modules and studied on weak co-
multiplication modules.
Definition 2.14 [7]: Let R be a commutative ring. An
R-module M is defined to be a weak co-
multiplication module if M doesn’t have any prime
submodule or for every prime sub module N of M,
N = (0 :M I) for same ideal I of R.
One can easily show that if M is a weak co-
multiplication module, then N = (0 :M Ann(N)) for
every prime submodule N of M.
We note that every co-multiplication module is weak
co-multiplication module. (see [19])
It should be mentioned that another dual notion of
weak multiplication module is defined in [19] as
second submodule.
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Definition 2.15 [19]: A submodule N of M is said to
be a second submodule if rN = N or rN = 0 for
each r R.
Definition 2.16 [19]: An R-module M is a weak co-
multiplication module if M does not have any
second submodule or for every second submodule or
for every second submodule S of M, we have
S = (0 :M I), where I is an ideal of R.
Remark in H. Ansari-Toroghy and F. Farshadifar
[19] it is clear that every co-multiplication R-module
is a weak co-multiplication R-module. However the
converse is not true in general.
For example, the Z-module Q (Here Q denotes the
field of rational numbers) is a weak co-multiplication
Z-module while it is not a co-multiplication
Z-module.
In H. Ansari-Toroghy and F. Farshadifar [19] M
is called weak co-multiplication module when for
each second submodule N of M, there is an ideal I of
R, such that N = (0 :M I) but here we are the term
weak co-multiplication module exclusively in sense
of [7] only.
Definition 2.17 [13]: Let R be a commutative ring.
An R-module L is said to be co-cyclic if L E (R|P)
for some maximal ideal P of R. (See [13] and [14])
Proposition 2.18 [8]: Every co-cyclic module over a
commutative complete Noetherian ring is a co-
multiplication module.
Proof: (See [8])
Definition 2.19 [15]: An R-module M is said to be
semi simple module (resp. co-semisimple module) in
case every submodule of M is the sum (resp.
intersection) of minimal (resp. maximal) submodule.
(Also See [16])
III. MAIN RESULTS
There is a large body of researches
concerning with multiplication modules. It is natural
to ask the following question: to what extent does the
dual of their results hold for co-multiplication
modules. Similar questions arise for weak
multiplication modules and weak co-multiplication
modules. The purpose of this paper is to obtain more
information about this class of modules.
Proposition 3.1 [8]: every co-cyclic module over a
commutative complete Noetherian ring is a co-
multiplication module.
Remark 3.2: Since every co-multiplication module
is weak co-multiplication module therefore every co-
cyclic module over a commutative complete
Noetherian ring is a weak co-multiplication module.
Proposition 3.3 [15]: Let M be an R-module and if
M is co-semi simple module such that Ann R (N) ≠
Ann R (M) for every maximal submodule N of M,
then M is co-multiplication module.
Remark 3.4: Since every co-multiplication module
is weak co-multiplication module. Therefore, let M
be an R-module and if M is co-semisimple module
such that Ann (N) ≠ Ann (M) for every maximal
submodule N of M then M is weak co-multiplication
module.
Proposition 3.5 [15]: Let M be an R-module if R is
Noetherian ring then every injective multiplication
R-module is co-multiplication module.
Remark 3.6: Since every co-multiplication module is
weak co-multiplication module. Therefore, Let M be
an R-module if R is Noetherian ring then every
injective multiplication R-module is weak co-
multiplication module.
As multiplication modules have been studied by
many mathematicians from different point of view. A
most suitable reference is Z. A. EI-Bast and P. F.
Smith [2]. Particularly in H. Ansari - Toroghy and F.
Farshadifar [8] have introduced co-multiplication
modules as a dual to multiplication modules. Under
various conditions co-multiplication modules are
cyclic. (See [18])
Lemma 3.7 [18]:
(1) If a submodule N of M equals (0 :M I) for same
ideal I of R then (N : M) = (Ann (M) :R I)
(2) M is a weak co-multiplication R-module if and
only if it is a weak co-multiplication ( R/Ann
(M) ) - module.
(3) Suppose that M is an R-module and R = R1×R2,
where R1 and R2 are non- trivial rings. Then M =
M1 M2 where M1 is an R1 module and M2 is an
R2-module. Also in this case M is weak co-
multiplication module if and only if both M1 and
M2 are weak co-multiplication modules.
Proof: (see [18])
Thus the results for weak co-multiplication
modules follow the similar proof of co-multiplication
modules.
According to above lemma 3.7, when M is a co-
multiplication module, it may be reduced to a faithful
co-multiplication module. (See [18])
Similarly when M is a weak co-multiplication
module, it may be reduced to a faithful weak co-
multiplication module.
Theorem 3.8 [18]: If M is a faithful weak co-
multiplication R-module with a maximal submodule
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N and R is a reduced ring with decomposition as a
finite direct product of indecomposable ring, then M
R and R is semi-simple.
Proof: See [18]
Corollary 3.9 [18]: Assume that M is a weak co-
multiplication module having a maximal submodule
(for example, if M is finitely generated) and Y =
Ann (M). Then Y is a prime ideal if and only if Y is a
maximal ideal and M R / Y is a simple module.
Corollary 3.10 [18]: If M is a finitely generated co-
multiplication module with Ann (M) a radical ideal,
then M is cyclic and R / Ann (M) is a semisimple
ring.
Proof: see [18].
Definition 3.11 [18]: A chained ring is a ring in
which every two ideals are comparable. For example,
localization of Z at any prime ideal or more generally
every valuation domain is a chained ring.
Lemma 3.12 [18]: If R is a chained ring and M is a
co-multiplication module having a maximal
submodule N, then M is cyclic.
Proof: see [18].
Remark 3.13: Since every co-multiplication module
is weak co-multiplication module. (see [19]), so for
the above results, under the various conditions, weak
co-multiplication modules are also cyclic.
Thus since every co-multiplication module
is weak co-multiplication module, so for the above
results, under the various conditions, weak co-
multiplication modules are also cyclic.
IV. Acknowledgement
This research was carried out by grant received
from the funding agency Chhattisgarh Council of
Science and Technology (CCOST) Raipur 492001
INDIA. First and second author would like to thanks
to CCOST Raipur for the financial support.
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