Question

Show that any noetherian module M is hopfian.

Answer 1

Let $\alpha: M \to M$ be surjective and $M$ be noetherian. The ascending chain $\ker \alpha \subseteq \ker \alpha 2 \subseteq \cdots$ must stop, so $\ker \alpha^i = \ker \alpha^{i+1}$ for some $i$. If $\alpha(m) = 0$, write $m = \alpha^i(m')$ for some $m' \in M$. Then

$$0 = \alpha \left(\alpha^ {i} \left(m ^ {\prime}\right)\right) = \alpha^ {i + 1} \left(m ^ {\prime}\right)$$

implies that $0 = \alpha^i(m') = m$, so $\alpha \in \operatorname{Aut}(M)$.