Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 1E / b

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 andThe 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.