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

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.
This question was merged with $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$ because it is an exact duplicate of that question.