Let $G=Z_3$ be acting on $R=\mathbb{C}[x,y]$ such that $g.x=\omega x$ and $g.y={\omega}^2y$, where $\omega$ is the $3rd$ root of unity. The action satisfies $g.(uv)=(g.u)(g.v)$ and $g.(u+v)=g.u+g.v$ for all $u,v \in R$ and $g\in G$. Let $R^G=\{r\in R \mid g.r=r\ \forall g\in G\}$, then show that $\frac{R^G}{< x^3,y^3 > }$ is a $\mathbb{C}$-vector space and find its dimension.
I tried to work out with $R=C[x]$ and the space $R'=\frac{R^G}{< x^3 > }$. After some rough work I arrived at the conclusion that it has dimension 1 over $\mathbb{C}$. Is it correct?
Can anyone give me some hints for the problem?