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Abstract Algebra: An Introduction
Found in: Page 149
Abstract Algebra: An Introduction

Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

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Short Answer

(a): Show that 4,6=2 in , where 4,6 is the ideal generated by 4 and 6 and 2 is the principal ideal generated by 2.

( b): Show that 6,9,15=3 in .

a. It can be proved that 4,6=2 in .

b. It can be proved 6,9,15=3 in .

See the step by step solution

Step by Step Solution

Lemma used

Let R be a commutative ring.

Let c1,c2,....,cndenote the ideal generated by c1,c2,......,cn in R

If J is an ideal of R and c1,c2,.....,cnJ , then localid="1648707689871" c1,c2,.....cnJ

Proving 4,62

It is clear that 4,62 in .

So, we can conclude 4,62 .

Proving 24,6

We can write 2=-1·4+1·6 .

This implies 24,6 .

Hence,24,6 .



Hence, 2=4,6.

Prove  6,9,15⊆3

Let R be a commutative ring.

Let c1,c2,...,cn denote the ideal generated by c1,c2,...cn in R

If J is an ideal of R and c1,c2,...,cnJ , then c1,c2,...,cnJ.

Proving 6,9,153

It is clear 4,62 in since 6, 9, 15 are multiples of 3.

So, we can conclude 6,9,153 .

Proving 36,9,15

We can write 3=-1·6+1·9+0·15 .

This implies 36,9,15 .

Hence,36,9,15 .



Hence, 3=6,9,15 .

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