Lemma 10.31.10. Any surjective endomorphism of a Noetherian ring is an isomorphism.

Proof. If $f : R \to R$ were such an endomorphism but not injective, then

$\mathop{\mathrm{Ker}}(f) \subset \mathop{\mathrm{Ker}}(f \circ f) \subset \mathop{\mathrm{Ker}}(f \circ f \circ f) \subset \ldots$

would be a strictly increasing chain of ideals. $\square$

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