Ideal is proper Ideal

Question

Let $R=\mathbb{Z}[X,Y]$, how can i show that $I=\langle 1-X,Y+X \rangle \neq R$?

I suppose that $1 \not\in I$, but I'm having a hard time showing that.

Answer 1

Way 1. Check that every polynomial in the ideal has $(-1,1)\in\Bbb Z^2$ as a root. So, $1\notin I$.

Way 2. Check $I = \langle X-1, Y-1 \rangle$. You may find it easier to do it with this. A simpler exercise could be to show that $\langle X,Y\rangle$ is a proper ideal. That should give you the idea.