Page 187 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 18: Integral Domains

CONTENTS
Objectives

Introduction
18.1 Integral Domains
18.2 Field
18.3 Summary
18.4 Keywords

18.5 Review Questions
18.6 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss whether an algebraic system is an integral domain or not

Explain the characteristic of any ring

Describe whether an algebraic system is a field or not

Introduction

In the earlier units, we have introduced you to rings, and then to special rings whose speciality
lay in the properties of their multiplication. In this unit, we will introduce you to yet another
type of ring, namely, an integral domain. You will see that an integral domain is a ring with
identity in which the product of two non-zero elements is again a non-zero element. We will
discuss various properties of such rings.
Next, we will look at rings like Q, R, C, and Z,, (where p is a prime number). In these rings, the
non-zero elements form an abelian group under multiplication. Such rings are called fields.
These structures are very useful, one reason being that we can “divide” in them.
Related to integral domains and fields are certain special ideals called prime ideals and maximal
ideals. In this unit, we will also discuss them and their corresponding quotient rings.

18.1 Integral Domains

You know that the product of two non-zero integers is a non-zero integer, i.e., if m, n  Z such.
0
that m  0, n  0, then mn  0. Now consider the ring Z . We find that 2  0 and 3  , yet
6
2 . 3  0. So, we find that the product of the non-zero elements 2 and 3 in Z is zero.
6
As you will soon realise, this shows that 2 (and 3) is a zero divisor, i.e., 0 is divisible by 2
(and 3 ).

So, let us see what a zero divisor is.

180 LOVELY PROFESSIONAL UNIVERSITY