# Isomorphism between quotient rings of $\mathbb{Z}[x,y]$

I need to find the condition on $m,n\in\mathbb{Z}^+$ under which the following ring isomorphism holds: $$\mathbb{Z}[x,y]/(x^2-y^n)\cong\mathbb{Z}[x,y]/(x^2-y^m).$$

My strategy is to first find a homomorphism $$h:\mathbb{Z}[x,y]\rightarrow\mathbb{Z}[x,y]/(x^2-y^m)$$ and then calculate the kernel of $h$.

To achieve this, I furthermore try to identify the isomorphism between $\mathbb{Z}[x,y]$ and itself, which I guess is $$f:p(x,y)\mapsto p(ax+by,cx+dy)$$

where $ad-bc=\pm 1,a,b,c,d\in\mathbb{Z}$.

Then $f$ induces a homomorphism $h$. But from here I failed to move on.

I believe there is some better idea, can anyone help?

Updated:

It should be isomorphism between quotient rings, not groups. Very sorry for such mistake.

• I don't think such an isomorphism holds whenever $n\neq m$. – Potato Jun 24 '13 at 7:50
• @Potato, it is a problem from a formal examination, I don't think it could be wrong, at least I want to try.. – hxhxhx88 Jun 24 '13 at 8:00
• Yes, and I think the point is to show that no isomorphisms can exist for distinct $n$ and $m$. – Potato Jun 24 '13 at 8:01