Is $\Bbb Z_4 [X]$ an integral domain?

Question

Is $\Bbb Z_4 [X]$ an integral domain? Explain why you cannot use the fact that if a commutative ring $R$ is an integral domain, then $R[X]$ is an integral domain.

If $P,Q \in \Bbb Z_4[X]$, then $\Bbb Z_4[X]$ is an integral domain iff $PQ=0 \implies P=0$ or $Q=0$.

How can I verify this here? Should I just consider polynomials in $\Bbb Z_4 [X]$ and check cases?

Answer 1

$\mathbb{Z}_4$ isn't even an integral domain: $2\cdot 2 = 0.$