Mathematics Asked by BehavingEarth on December 10, 2021
I am using Atiyah MacDonald to study commutative algebra and in Chapter 9, it says the following:
If you look at the paragraph after the proof of 9.1, you can see that to get unique factorization, we must also have that given a nonzero prime ideal $mathfrak p$ and $n neq m$, we have $mathfrak p^n neqmathfrak p^m$. I was not able to see why this was the case. Any help will be appreciated.
It suffices to show that the chain $$mathfrak psupseteqmathfrak p^2supseteqmathfrak p^3 supseteq cdots$$ is strictly descending.
Suppose instead that $mathfrak p^{n+1}=mathfrak p^n$ for some positive integer $n$.
Let $M=mathfrak p^n$, regarded as an $A$-module.
Since $A$ is a domain and $mathfrak pne 0$, we get $Mne 0$.
Since $A$ is Noetherian, $M$ is finitely generated.
From $mathfrak p^{n+1}=mathfrak p^n$ we get $mathfrak pM=M$, hence by Nakayama's lemma
$;;;;$ https://en.wikipedia.org/wiki/Nakayama%27s_lemma#Statement
we get $aM=0$ for some $ain A$ with $aequiv 1;(text{mod};mathfrak p)$.
But then $amathfrak p^n=0$, contradiction, since $ane 0$,$;mathfrak p^nne 0$, and $A$ is a domain.
Answered by quasi on December 10, 2021
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