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### Abstract Algebra: An Introduction

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

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

No, it is not always true.

See the step by step solution

## 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, $I=\left\{rc\overline{)r\in R}\right\}$ Then $I$ is an ideal.

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

## Counter example

Consider a commutative ring with identity $\mathrm{ℤ}$ .

Let $R=\mathrm{ℤ}$.

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

$\left(3\right)$ and $\left(-3\right)$ are two ideals generated by $3$ and $-3$, respectively, in $\mathrm{ℤ}$ .

## Justification

Here,

$\left(3\right)=\left(-3\right)$ . But $3\ne -3$ .

That is, $\left(a\right)=\left(b\right)$ , but $a\ne b$ .

## Conclusion

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