Page 203 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 20: Principal Ideal Domains

CONTENTS
Objectives

Introduction
20.1 Euclidean Domain
20.2 Principal Ideal Domain (PID)
20.3 Summary
20.4 Keywords

20.5 Review Questions
20.6 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the Principal Ideal Domains

Describe theorem related to Principal Ideal Domains

Introduction

In the last unit, you have studied about field and integer domain. In this unit, you will study
about Principal Ideal Domains.

20.1 Euclidean Domain

In earlier classes you have seen that Z and F[x] satisfy a division algorithm. There are many
other domains that have this property. Here we will introduce you to them and discuss some of
their properties. Let us start with a definition.

Definition: Let R be an integral domain. We say that a function d : R \ (0)  NU (0) is a Euclidean
valuation on R if the following conditions are satisfied:

(i) d(a)  d (ab)  a, b  R \ {0}, and
(ii) for any a, b  k, b  0 3 q, r  R such that
a = bq+r, where r = 0 or d(r) < d(b).

And then R is called a Euclidean domain.
Thus, a domain on which we can define a Euclidean valuation is a Euclidean domain.
Let us consider an example.

Example: Show that Z is a Euclidean domain.
Solution: Define, d : Z  N{0} : d(n) = |n|.

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