Page 147 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 14: Rings

CONTENTS
Objectives

Introduction
14.1 What is a Ring?
14.2 Elementary Properties
14.3 Two Types of Rings
14.4 Summary

14.5 Keywords
14.6 Review Questions
14.7 Further Readings

Objectives

After studying this unit, you will be able to:
Define and give examples of rings

Discuss some elementary properties of rings from the defining axioms of a ring

Define and give examples of commutative rings, rings with identity and commutative

rings with identity

Introduction

With this unit, we start the study of algebraic system with two binary operations satisfying
certain properties. Z, Q and R are examples of such a system, which we shall call a ring.
Now, you know that both addition and multiplication are binary operations on Z. Further, Z is
an abelian group under addition. Though it is not a group under multiplication, multiplication
is associative. Also, addition and multiplication are related by the distributive laws
a(b + c) = ab + nc, and (a + b)c = ac + bc

for all integers a, b and c. We generalise these very properties of the binary operations to define
a ring in general. This definition is given by the famous algebraist Emmy Noether.
After defining rings we will provide several examples of rings. You will also learn about
some propertics of rings that follow from the definition itself. Finally, we shall discuss certain
types of rings that are obtained when we impose more restrictions on the “multiplication” in
the ring.
As the contents suggest, this unit lays the foundation for the rest of this course. So make sure that
you have attained the following objectives before going to the next unit.

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