Artinian modules from Cohn's ring theory

[Math Amateur]

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:

Any help will be much appreciated ...

Peter

 

[fresh_42]

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 $coker(φ^n) = M / im(φ^n) , n ∈ℕ.$