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Richa Nandra, Lovely Professional University Unit 21: Unique Factorization Domains

Unit 21: Unique Factorization Domains Notes

CONTENTS
Objectives

Introduction
21.1 Unique Factorisation Domain (UFD)
21.2 Summary
21.3 Keyword
21.4 Review Questions

21.5 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss unique factorization domains

Explain theorems of UID

Introduction

In this unit, we shall look at special kinds of integral domains. These domains were mainly
studied with a view to develop number theory. Let us say a few introductory sentences about
them.

You saw that the division algorithm holds for F[x], where F is a field. You saw that it holds
for Z. Such integral domains are called Euclidean domains.
We shall look at some domains which are algebraically very similar to Z. These are the principal
ideal domains, so called because every ideal in them is principal.
Finally, we shall discuss domains in which every non-zero non-invertible element can be uniquely
factorised in a particular way. Such domains are very appropriately called unique factorisation
domains. While discussing them, we shall introduce you to irreducible elements of a domain.

While going through the unit, you will also see the relationship between Euclidean domains,
principal ideal domains and unique factorisation domains.

21.1 Unique Factorisation Domain (UFD)

Here we shall look at some details of a class of domains that include PDs.

Definition: We call an integral domain R a unique factorisation domain (UFD, in short) if every
non-zero element of R which is not a unit in R can be uniquely expressed as a product of a finite
number of irreducible elements of R.

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