Ring Theory And Its Basic Application

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.
(A3) There is an element 0 R satisfying a + 0 = a for all a in R.
(A4) For every a R there is an element b R such that a + b = 0.
(R2) R is associative under multiplication: that is, (a.b).c = a.(b.c), for all a, b, c 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
The work is organized as follows: chapter one discuses the introductory part of the work, chapter two presents the literature review of the related works, chapter three describes the theory and its formulas, chapter four discusses result, chapter five is on summary of findings, conclusion and recommendation.