Show that if $P_n$ is a descending chain of submodules in $N$, then so is $f^{-1} (P_n)$ in $M$

Question

Given $f:M→N$ surjective, show that if $P_n$ is a descending chain of submodules in $N$, then $f^{-1} (P_n)$ is a descending chain of submodules in $M$.

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I'm a bit lost to be completely honest. The book I'm using assumes this as fact to prove that some module is Artinian. Any tips?

Answer 1

Note that taking the inverse preserves inclusions (if $A \subseteq B$, then $f^{-1}(A) \subseteq f^{-1}(B)$). This is a general property of functions, not module morphisms.

Lastly, the primage of a module is a module. This is not particularly hard to prove. Mainly observe that if $A$ is a submodule and $x,y \in f^{-1}(A)$, then $f(x+y) = f(x) + f(y) \in A$. Similar for scalar multiplication.