Page 195 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 19: The Field of Quotient Euclidean Domains

CONTENTS
Objectives

Introduction
19.1 Prime and Maximal Ideals
19.2 Field of Quotients
19.3 Summary
19.4 Keywords

19.5 Review Questions
19.6 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss whether an algebraic system is an integral domain or not

Define and identify prime ideals and maximal ideals

Prove and use simple properties of integral domains and fields

Construct or identify the field of quotients of an integral domain

Introduction

Finally, we shall see how to construct the smallest field that contains a given integral domain.
This is essentially the way that Q is constructed from Z. We call such a field the field of quotients
of the corresponding integral domain.
In this unit, we have tried to introduce you to a lot of new concepts. You may need some time to
grasp them. Take as much time as you need. But by the time you finish it, make sure that you
have attained the knowledge of following topics.

19.1 Prime and Maximal Ideals

In ‘Z’ we know that if p is a prime number and p divides the product of two integers a and b, then
either p divides a or p divides b. In other words, if ab  pZ, then either a  pZ or b  pZ. Because
of this property we say that pZ is a prime ideal, a term we will define now.

Definition: A proper ideal P of a ring R is called a prime ideal of R if whenever ab  P for a, b 
R, then either a  P or b  P.
You can see that {0} is a prime ideal of Z because ab  {0}  a  {0} or b  {0}, where a,b  Z.

188 LOVELY PROFESSIONAL UNIVERSITY