Thread: Artinian Modules - Cohn Exercise 3, Section 2.2, page 65

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I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ...

I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ...

I need help to get started on Exercise 3, Section 2.2, page 65 ...

Exercise 3 (Section 2.2, page 65) reads as follows:
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Any help will be much appreciated ...

Peter
 

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The same exercise is in my book (see your other thread). There Atiyah, Macdonald give the following hints:
Let ##φ : M → M## be the endomorphism. To prove the Noetherian case consider the submodules ##ker(φ^n)## and in the Artian case the quotient modules ##coker(φ^n) = M / im(φ^n) , n ∈ℕ.##
 
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