Here $\mathbb{F}$ is an arbitrary field. $$\psi :\mathbb{F}[X,Y]\rightarrow\mathbb{F}[X]\times\mathbb{F}[X];\\ f\mapsto (f \ \text{mod} (X+Y),f\ \text{mod} (X-Y))$$Is a ring homomorphism with kernel $(X^2-Y^2)$, so $\mathbb{F}[X,Y]/(X^2-Y^2)$ is isomorphic to a subring of $\mathbb{F}[X]^2$.

What is the form of this subring?

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    $\begingroup$ $F[r,s]/(rs)$. Send $r=X+Y$ and $s=X-Y$. $\endgroup$ – user647486 Apr 7 at 11:04

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