# Answer to Question #17646 in Abstract Algebra for Hym@n B@ss

Question #17646

Show that any artinian module M is cohopfian.

Expert's answer

Let

*α*:*M → M*be injective, and*M*be artinian. The descending chain im(*α*)*⊇**im(**α**^*2)*⊇**· · ·*muststabilize, so im(*α**^i*) = im(*α**^i*+1) for some*i*. For any*m**∈**M*, we have*α**^i*(*m*) =*α**^i*+1(*m'*)for some*m'**∈**M*. But then*α**^i*(*m −**α*(*m'*)) = 0 implies that*m*=*α*(*m'*), so*α**∈**Aut**R*(*M*).
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