We know that if $f\in k[X_1,{...},X_n]$ is quasi-homogeneous polynomial and $R :=k[X_1,{...},X_n]/(f)$, then any minimal generating set of $\operatorname{Der}_k(R)$ contains the Euler derivation. Is the same thing true for any positive graded domain?
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$\begingroup$ I may be misunderstanding your question. If you take $R=k[x]$, the derivations are generated by $d/dx$ and the Euler derivation is $x\cdot d/dx$. $\endgroup$– MohanCommented Dec 9, 2022 at 18:37
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