Is every non-trivial ideal in a commutative ring is a principal ideal?

Question

I'm a bit lost... it seems every non-trivial ideal in a commutative ring is a principal ideal. but is it true? if not, could you pls give a counter example?

Answer 1

By non-trivial, do you an ideal that is not the whole ring itself? I'm going to proceed assuming that's what you mean.

If every ideal in a given integral domain $R$ is a principal ideal, then $R$ is a principal ideal domain, and then it's also a unique factorization domain. In $\mathbb{Z}[\sqrt{-2}]$, for example, every ideal is a principal ideal.

But now consider $\mathbb{Z}[\sqrt{-5}]$, which is neither a UFD nor a PID. $\langle 3 \rangle$ is a principal ideal but not a prime ideal. The ideal $\langle 3, 1 + \sqrt{-5} \rangle$, which consists of all numbers of the form $3a + b\sqrt{-5}$ with $\{a, b\} \in \mathbb{Z}[\sqrt{-5}]$ is a prime ideal but not a principal ideal.