Question

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.

Short answer

No, it is not always true.

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$\mathrm{\mathbb{Z}}$ .

Let $R=\mathrm{\mathbb{Z}}$.

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

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

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$b$ .