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This question is from notes of algebraic geometry from which I am self studying and I need help with it.

Question (a) Let A be a commutative ring and M be a noetherian A-module. Prove that A/ Ann(M) is a noetherian ring.

Work: M is noetherian means that every ascending chain of subsets of A will terminate and also that M is a finitely generated A-module. Ann(M) = { $x\in A|$ x.m =0 for all $m\in M$ }But I am not able to get any intuition on how to prove that A/Ann(M) is noetherian.

(b) If A is noetherian, then prove that any surjective hom $\phi:A\to A$ is an isomorphism.

Work: I have to prove that $\phi(a) =0 $ implies that a=0 , given A is surjective. But how should I proceed? Can you please give a hint?

user26857
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    These are two separate questions and should be asked in separate posts. The second question is already answered on MSE here. – KReiser Dec 15 '21 at 10:35
  • @KReiser In notes they are asked in same question in two parts (a) and (b) that's why I asked this way. –  Dec 15 '21 at 10:37
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    It turns out that both of your questions have been addressed here on MSE before, therefore I've marked your post as a duplicate of the relevant questions. – KReiser Dec 15 '21 at 10:43
  • @KReiser Got it! –  Dec 15 '21 at 10:51
  • https://math.stackexchange.com/questions/3355919/quotient-of-annihilator-module –  Jan 15 '22 at 13:59

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