Page 165 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 16: Ideals

CONTENTS
Objectives

Introduction
16.1 Quotient Rings
16.2 Summary
16.3 Keywords
16.4 Review Questions

16.5 Further Readings

Objectives

After studying this unit, you will be able to:

Discuss ideals of some familiar rings

Explain whether a subset of a ring is an ideal or not

Define and give examples of quotient rings

Introduction

In earlier unit, you have studied normal subgroups and the role that they play in group theory.
You saw that the most important reason for the existence of normal subgroups is that they allow
us to define quotient groups. In ring theory, we would like to define a similar concept, a
quotient ring. In this unit, we will discuss a class of subrings. These subrings are called ideals.
While exploring algebraic number theory, the 19th century mathematicians Dedekind, Kronecker
and others developed this concept. Let us see how we can use it to define a quotient ring.
Consider a ring (R, + , .) and a subring I of R. As (R, +) is an abelian group, the subgroup, I is
normal in (R, +), and hence the set R/I = ( a + 1 | a  R }, of all cosets of I in R, is group under the
binary operation + given by
(a + I) + (b + I) = (a + b) + I ..... (1)

for all a + I, b + I  R/I. We wish to define. on R/I so as to make R/I a ring. You may think that
the most natural way to do so is to define

(a + I) . (b + I) = a b + I  a + 1, b + I  R ..... (2)
But, is this well defined? Not always. For instance, consider the subring Z of R and the set of
cosets of Z in R. Now, since 1 = 1 – 0  Z , 1 + Z = 0 + Z.

Therefore, we must have
( 2 + Z) . (1 + Z) = ( 2 + Z) . (0 + Z), i.e,, 2 + Z = O + Z, i.e., 2  Z.

But this is a contradiction. Thus, our definition of multiplication is not valid for the set R/Z.

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