Solution
= Solution
Equality of constant terms is preserved by componentwise addition, subtraction, and multiplication, and $(1,1)$ belongs to the set, so $R$ is a subring. The projection is surjective and
$$
\ker p=\{(0,yg(y)):g\in\mathbb C[y]\}.
$$
The nonzero elements $(x,0)$ and $(0,y)$ have product zero, so $R$ is not an <integral> domain and hence not a PID. In fact $R\cong\mathbb C[x,y]/(xy)$, which also shows that it is Noetherian.