Question

Show that, if M_R is an artinian module, then M is cohopfian.

Answer 1

Assume, for the moment, that $g(M) \subset M$, $g(M) \neq M$. Since $g$ is an injection,

$g^{2}(M)\subset g(M),g^{2}(M)\neq g(M)$. Repeating this argument, we see that $g^{n + 1}(M)\subset g^n (M),g^{n + 1}(M)\neq g^n (M)$ for all $n$. Therefore, we have a strictly descending chain

$$M \supset g (M) \supset g ^ {2} (M) \supset \dots$$

contradicting the fact that $\mathbf{M}$ is an artinian module.