Torsion-free quotient of integer polynomial ring

Question

Consider the ring of polynomials $\mathbb{Z}[x,y]$ and let $I$ be the ideal $(xy,x+y)$. Is the quotient $\mathbb{Z}[x,y]/I$ torsion-free as a $\mathbb{Z}$-module? How does one approach this type of question in the more general case when $I$ is generated by several given polynomials?

Answer 1

The quotient ring is isomorphic to $\mathbb{Z}[x]/(x^2)$ (since we have $y=-x$ in the quotient, so that we may get rid of $y$ and write $xy=0$ as $x^2=0$), hence free of rank $2$ over $\mathbb{Z}$.