Is $\mathbb{Z}[X]/(I+J)$ integral domain

Question

Let $I=\left<X-2\right>, J=\left<X+2\right>$. Is $\mathbb{Z}[X]/(I+J)$ integral domain?

We have a theorem which says: $I\triangleleft P$ is prime ideal iff $P/I$ is domain.

So, in this case, we can prove if $I+J$ is prime ideal. How should I do it? Thank you.

Answer 1

As $X+2-(X-2)=4\in I+J$, we must have

$$(2+I+J)(2+I+J)=(4+I+J)=(0+I+J)$$

However, $2+I+J\neq 0+I+J$, as $2\notin I+J$. So, it is not an integral domain.

Edit: Why $2\notin I+J$? Suppose it is, then there are $p,q\in\mathbb{Z}[X]$ such that

$$p(X)(X+2)+q(X)(X-2)=2$$

Now, $p(2)(2+2)+q(2)(2-2)=2$, so $4p(2)=2$, contradiction.