RING THEORY AND ITS BASIC APPLICATION
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Ring theory is one of the parts of the abstract algebra that has been comprehensively utilized in images. Be that as it may, ring theory has not been connected with picture segmentation. In this paper, we propose another list of likeness among images utilizing - rings and the entropy function. This new file was connected as another ceasing standard to the Mean Shift Iterative Algorithm with the objective to achieve a superior segmentation. An investigation on the execution of the algorithm with this new halting standard is completed. Though ring theory and class theory at first pursued diverse bearings it turned out during the 1970s – that the study of functor classifications additionally uncovers new angles for module theory.
TABLE OF CONTENTS
COVER PAGE
TITLE PAGE
APPROVAL PAGE
DEDICATION
ACKNOWELDGEMENT
ABSTRACT
CHAPTER ONE
- INTRODUCTION
- BACKGROUND OF THE PROJECT
- AIM OF THE STUDY
- SCOPE OF THE STUDY
- APPLICATION OF THE STUDY
- BENEFIT OF RING THEORY
- RESEARCH QUESTION
- DEFINITION AND ELEMENTARY PROPERTIES OF RINGS
- RESEARCH METHODOLOGY
- PROJECT ORGANISATION
CHAPTER TWO
LITERATURE REVIEW
- OVERVIEW OF THE STUDY
- TYPES OF RING
- HISTORICAL BACKGROUND OF RING THEORY
- REVIEW OF RELATED STUDIES
CHAPTER THREE
METHODOLOGY
- INTRODUCTION
- RING IN CRYPTOGRAPHY
- RING THEORY IN THE SEGMENTATION OF DIGITAL IMAGES
CHAPTER FOUR
4.0 RESULT AND DISCUSSION
CHAPTER FIVE
- CONCLUSION
- REFERENCES
CHAPTER ONE
1.0 INTRODUCTION
1.1 BACKGROUND OF THE STUDY
In mathematics, a ring is an algebraic structure comprising of a set together with two binary operations for the most part called addition and multiplication, where the set is an abelian bunch under addition (called the additive gathering of the ring) and a monoid under multiplication to such an extent that multiplication disseminates over addition. As such the ring axioms necessitate that addition is commutative, addition and multiplication are cooperative, multiplication circulates over addition, every component in the set has an additive inverse, and there exists an additive personality. A standout amongst the most well-known examples of a ring is the arrangement of whole numbers supplied with its regular operations of addition and multiplication.
The part of mathematics that reviews rings is known as ring theory. Ring theorists study properties basic to both well-known scientific structures, for example, whole numbers and polynomials, and to the a lot less outstanding numerical structures that additionally fulfill the axioms of ring theory. The universality of rings makes them a focal sorting out guideline of contemporary mathematics.
Ring theory might be utilized to comprehend major physical laws, for example, those basic exceptional relativity and symmetry marvels in sub-atomic science.
Ring theory is commonly seen as a subject in Pure Mathematics. This implies it is a subject of natural magnificence. In any case, the possibility of a ring is fundamental to the point that it is additionally crucial in numerous utilizations of Mathematics. Without a doubt it is fundamental to the point that a lot of other essential apparatuses of Applied Mathematics are worked from it. For example, the vital idea of linearity, and straight algebra, which is a down to earth need in Physics, Chemistry, Biology, Finance, Economics, Engineering, etc, is based on the thought of a vector space, which is a unique sort of ring module. Ring theory seems to have been among the most loved subjects of the absolute most compelling Scientists of the twentieth century, for example, Emmy Noether; and Alfred Goldie. In any case, maybe more essential than any of these focuses is that ring theory is a center piece of the subject of Algebra, which frames the language inside which present day Science can be put on its firmest conceivable balance.
1.2 AIM OF THE STUDY
The main aim of this work is to carry out a study on ring theory and its general application.
1.3 SCOPE OF THE STUDY
This study is on Ring theory and its application. This work studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications.
1.4 APPLICATION OF THE STUDY
Ring theory can be understood at a moderate level by high-school level students both in mathematics and computer department, and in fact well enough by interested undergraduate students for them to carry out a research on ring theory.
1.5 BENEFIT OF RING THEORY
Ring theory provides the basic framework for understanding how to manipulate matrix addition and multiplication together. The theory of modules over rings generalizes that of vector spaces over fields.
1.6 RESEARCH QUESTION
i. What is ring in ring theory?
ii. What are ideals in ring theory?
iii. What is the function of a ring?
iv. What is ring and field?
1.7 DEFINITION AND ELEMENTARY PROPERTIES OF RINGS
A ring is an ordered triple (R, +,.) consisting of a non-empty set R and two binary operations on R called addition (+) and multiplication (.), satisfying the following properties:
(R1) (R, +) is an abelian group, that is,
(A1) a + (b + c) = (a + b)+ c for all a, b, c in R.
(A2) a + b = b + a for all a, b in R.
∈
∈ ∈
∈
(R3) Multiplication is distributive (on both sides) over addition; that is, a.(b + c) =
∈
a.b + a.c and (a + b).c = a.c + b.c , for all a, b, c R. (The two distributive laws are respectively called the left distributive law and the right distributive law.)
1.8 RESEARCH METHODOLOGY
In the course of carrying this study, numerous sources were used which most of them are by visiting libraries, consulting journal and news papers and online research which Google was the major source that was used.
1.9 STRUCTURE OF THE STUDY
CHAPTER FIVE
5.1 CONCLUSIONS
In this work, a new stopping criterion, for the iterative MSHi, based on the Ring Theory was proposed. The new stopping criterion establishes a new measure for the comparison of two images based on the use of the entropy concept and the spatial information. The quotient space was defined using the equivalent classes of images, to be able of selecting any element of the class. Through the obtained theoretical and practical results, it was possible to prove that the new stopping criterion had very good performance in the algorithm MSHi, and was more stable that the old criterion.
A new public key cryptographic system using group rings. A group ring RG is chosen. A unit of this group ring is generated which is then used as the public key. The method of generation of the unit or public key is such that the inverse of the unit which is then the private key, cannot be obtained from the public key. The plaintext to be transmitted is converted into a (unique) group ring element and the chosen unit (public key) is used to generate the ciphertext. The ciphertext is transmitted and only the person holding the private key, which is the inverse of the unit, can decrypt the ciphertext.
- Two or more chosen units of the same or of different types may be combined to form a new unit which is not of any of the types of the constituent units. This new unit is then used as the public key and is not of any known type. The private key is kept secret and is the combination in a certain order of the inverses of the constituent units.
- The group rings used may be commutative or non-commutative. Known units in the non-commutative case may be combined together (or with commutative units) to give a new unit not of the same type as any of the constituents. This new unit is then the public key and the private key is obtained from the inverses of the constituents; the public key would not reveal either the inverses of the constituents nor the types of the constituents.
- Given a unit of a particular type, a power of this unit and products of such powers may be used as the public key. Using a power of a unit also introduces the difficulty of the discrete logarithm problem so that knowing the power of the unit is not enough to know the unit itself. This power of a unit may also be combined with other keys or units which have been generated according to this or previous methods to give a new unit and hence a new public key and a new private key; this new system has both the difficulty of trying to find the inverse of a unit and the difficulty of the discrete logarithm problem.
- A method exists to disguise the size of the units or keys. A method exists to deal with large numbers.
- These units or keys generated may be combined with existing public key cryptosystems such as RSA. The new combined systems have more secu- rity than each of the constituent systems and will not be of either type. Thus, for example, a system combining group ring public key and RSA public key will have more security than the group ring public key system or the RSA system and will not be of either type.
- In many cases these group ring public key cryptosystems can be combined with coding and error-correcting codes to give one system with both public key and coding all in one system.
- Many variations and examples are possible.
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