# Surjective endomorphisms of Noetherian modules are isomorphisms.

I'm trying to solve this question:

I didn't understand why the hint is true and how to apply it. I really need help, because it's my first question on this subject and my experience on this field is zero.

I need some help.

Thanks a lot

• Already proved here. – user89712 Nov 17 '13 at 21:36

For any $m\in M$, if you have $u(m)=0$, then you have $$(u\circ u)(m)=u(u(m))=u(0)=0.$$ Therefore $\ker(u)\subseteq\ker(u\circ u)$. By the same argument, you have $$\ker(u)\subseteq\ker(u\circ u)\subseteq\ker(u\circ u\circ u)\subseteq\cdots$$ which should look familiar if you have just read a chapter on Noetherian modules.