Why is $ℤ[X]/(X^2-1)$ not isomorphic with $ℤ×ℤ$?

Question

Why is $ℤ[X]/(X^2-1)$ not isomorphic with $ℤ ×ℤ$ ?

I understand why this is true in the case of $\mathbb{Q}$. But $(X-1)+(X+1) ≠ℤ[X]$, so therefore I can't use the same reasoning. I don't see how I can proof that there can't be an isomorphism.