Question

Question 5:Let l be an ideal in an integral domainR. Is it true thatR/Iis also an integral domain.

Short answer

No, this may not be true thatR/I is an integral domain.

Step-by-step solution

Give a Counter example

We have to prove that statement may not always be true with the help of a counter example.

Consider an integral domain:

Let $\mathrm{\mathbb{Z}}$is an integral domain.

It is clear that $6\mathrm{\mathbb{Z}}$is an ideal in$\mathrm{\mathbb{Z}}$ .

Check the statement

But $\mathrm{\mathbb{Z}}/6\mathrm{\mathbb{Z}}\cong {\mathrm{\mathbb{Z}}}_6$and${\mathrm{\mathbb{Z}}}_6$ is not an integral domain.

Hence, $\mathrm{\mathbb{Z}}/6\mathrm{\mathbb{Z}}$is not an integral domain.

Thus,$R/I$ is not an integral domain.