Lemma 114.5.5. Let $k$ be a field. Let $F, G \in k[X, Y]$ be homogeneous of degrees $d, e$. Assume $F, G$ relatively prime. Then multiplication by $G$ is injective on $S = k[X, Y]/(F)$.

**Proof.**
This is one way to define “relatively prime”. If you have another definition, then you can show it is equivalent to this one.
$\square$

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