kernel of homomorphism $\mathbb{C}[x,y] \to \mathbb{C}[t]$ but in general case

Question

Let $f:\mathbb{C}[x,y] \to \mathbb{C}[t]$ be a homomorphism that is identity on $\mathbb{C}$ and sends $x\to x(t),y \to y(t)$ and such that $x(t),y(t)$ aren't both constant. Prove $\ker(f)$ is a principal ideal.

Answer 1

The kernel is a prime ideal. The source has Krull dimension two, while the target is a subring of $\mathbb C[t]$ which is bigger than $\mathbb C$, and so has Krull dimension one. Thus the kernel has height one, i.e. it is minimal with respect to being non-zero. Thus it is principal (since $\mathbb C[x,y]$ is a UFD).