Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

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Short Answer

If $R$ is a commutative ring with identity and $(a)$ and $(b)$ are principal ideals such that $(a) = (b)$ , is it true that $a = b$ ? Justify your answer.

No, it is not always true.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Definition

Let $R$ be a commutative ring with identity, $c \in R$ , and $I$ the set of all multiples of $c$ in $R$ : that is, $I = \{r c r \in R\}$ Then $I$ is an ideal.

Here, the ideal $I$ is called the principal ideal generated by $c$ and is denoted by $(c)$

Counter example

Consider a commutative ring with identity $ℤ$ .

Let $R = ℤ$ .

Let $a = 3$ and $b = - 3$ .

$(3)$ and $(- 3)$ are two ideals generated by $3$ and $- 3$ , respectively, in $ℤ$ .

Justification

Here,

$(3) = (- 3)$ . But $3 \neq - 3$ .

That is, $(a) = (b)$ , but $a \neq b$ .

Conclusion

Hence, if $R$ is a commutative ring with identity and $(a)$ and $(b)$ are principal ideals such that, $(a) = (b)$ then $a$ may not be equal to $b$ .

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Abstract Algebra: An Introduction

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Short Answer

If $R$ is a commutative ring with identity and $(a)$ and $(b)$ are principal ideals such that $(a) = (b)$ , is it true that $a = b$ ? Justify your answer.

Step by Step Solution

Definition

Counter example

Justification

Conclusion

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Abstract Algebra: An Introduction

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Short Answer

If Ris a commutative ring with identity and a and b are principal ideals such that a=b, is it true that a=b? Justify your answer.

Step by Step Solution

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Counter example

Justification

Conclusion

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If $R$ is a commutative ring with identity and $(a)$ and $(b)$ are principal ideals such that $(a) = (b)$ , is it true that $a = b$ ? Justify your answer.