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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