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Let $h \in \mathbb{C}[x]$ with $\deg(h)=d \geq 1$ and let $R_{h,y}:=\mathbb{C}+\langle h,y \rangle \subseteq \mathbb{C}[x,y]$.

It is known that subalgebras of $\mathbb{C}[x]$ are finitely generated, see this, therefore, $\mathbb{C}+\langle h \rangle$ is finitely generated, and it is generated by $\{h,xh,\ldots,x^{d-1}h\}$. For example, for $h=x^2$ we obtain $\mathbb{C}+\langle x^2 \rangle=\mathbb{C}[x^2,x^3]$, so a finite set of generators is, for example $\{x^2,x^3\}$.

There are non-finitely generated subalgebras of $\mathbb{C}[x,y]$, see this.

Is $R_h$ finitely generated? Probably not? It is not true that $R_{h,y}=\mathbb{C}[y, x^iy^jh]_{0 \leq i+j \leq d-1}$, since $xy \in \mathbb{C}+\langle h,y \rangle$, but $xy \notin \mathbb{C}[y, x^iy^jh]_{0 \leq i+j \leq d-1}$.

Edit: Could one please present a finite list of generators, if exists.

Thank you very much!

user237522
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  • $\mathbb{C}[h,y]=A\subset R_{h,y}\subset \mathbb{C}[x,y]=S$. $S$ is a finite type module over $A$ and thus, so is $R_{h,y}$ hence finitely generated. – Mohan Jan 25 '21 at 23:14
  • Thank you very much! I am not able to find a finite set of generators for $R_{h,y}$ over $\mathbb{C}$. $R_{h,y}$ includes, for example, ${x^ny}_{n \in \mathbb{N}}$. – user237522 Jan 26 '21 at 07:02

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