Is $\mathbb{Z}_3[x] / \langle x^2+x+1 \rangle $ an integral domain?

Question

I can prove that it is not a field. But that does not imply that it is not an integral domain either.So I am facing difficulty in solving this. Thank you for any help in advance.

Answer 1

$(x-1)^2 = x^2 +x + 1$ over $\Bbb Z_3$, so $y:=[x-1]$ is nonzero in the quotient with $y^2= 0$. So it's not an integral domain.

In general, note that if $p(x)$ is a polynomial over a field $F$, and it's not irreducible, so $p(x)=q(x)r(x)$, for non trivial polynomials then the quotient $F[x]{/}\langle p(x) \rangle$ has $[q(x)]$ and $r(x)]$ as zero-divisors. So $F[x]{/}\langle p(x) \rangle$ has two options: a field (if $p(x)$ is irreducible), or "plain" commutative unitary ring if $p(x)$ is not.