Question

Show that any artinian module M is cohopfian.

Answer 1

Let α : M → M be injective, and Mbe 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 ∈ AutR(M).