To check whether given ring is principal ideal domain?

Question

Is $\mathbb{Z}_6[x]$ is principal ideal domain ?

no,

we know that polynomial ring $\mathbb{F}[x]$ is field iff $\mathbb{F}$ is field

here $\mathbb{Z}_6$ is not field.

consider ideal $\langle{x,2}\rangle$ which is not principal ideal in $\mathbb{Z}_6[x]$ hence it is not PID.

please correct me if i am wrong.

Answer 1

$\mathbb{Z}_6[x]$ is not a principal ideal domain because it is not even a domain.