Will $Z[X] / (x^2 +1)$ be a PID?
My Try : It is not a PID. If $R$ = $Z[X] / (x^2 +1)$ were a PID then $2$ will be an irreducible element in $R$ and $R/(2)$ is a field. But $x$ will not have an inverse element in $R/(2)$. contradiction
Have I gone wrong anywhere? Please correct me if I have.
Thank You in Advance.