Past exam of the commutative algebra course of the University of Oxford / 2023 / 2 / c

For $n\ge 1$, let $K_n=\ker ({\varphi }^n)$, where the power denotes repeated composition of the ring endomorphism. These ideals form an ascending chain:The ascending chain condition for a Noetherian ring gives $K_N=K_{N+1}$ for some $N\ge 1$. Take $a\in \ker \varphi$. Since $\varphi$ is surjective, its iterate ${\varphi }^N$ is surjective, so choose $b\in R$ with ${\varphi }^N(b)=a$. Thenso $b\in K_{N+1}=K_N$. Therefore $a={\varphi }^N(b)=0$.

Thus the kernel of a ring homomorphism $\varphi$ is zero. This proves that a surjective endomorphism of a Noetherian ring is an automorphism, and in particular