Lemma 10.62.1. Let $R$ be a Noetherian ring, and let $M$ be a finite $R$-module. There exists a filtration by $R$-submodules

$0 = M_0 \subset M_1 \subset \ldots \subset M_ n = M$

such that each quotient $M_ i/M_{i-1}$ is isomorphic to $R/\mathfrak p_ i$ for some prime ideal $\mathfrak p_ i$ of $R$.

First proof. By Lemma 10.5.4 it suffices to do the case $M = R/I$ for some ideal $I$. Consider the set $S$ of ideals $J$ such that the lemma does not hold for the module $R/J$, and order it by inclusion. To arrive at a contradiction, assume that $S$ is not empty. Because $R$ is Noetherian, $S$ has a maximal element $J$. By definition of $S$, the ideal $J$ cannot be prime. Pick $a, b\in R$ such that $ab \in J$, but neither $a \in J$ nor $b\in J$. Consider the filtration $0 \subset aR/(J \cap aR) \subset R/J$. Note that both the submodule $aR/(J \cap aR)$ and the quotient module $(R/J)/(aR/(J \cap aR))$ are cyclic modules; write them as $R/J'$ and $R/J''$ so we have a short exact sequence $0 \to R/J' \to R/J \to R/J'' \to 0$. The inclusion $J \subset J'$ is strict as $b \in J'$ and the inclusion $J \subset J''$ is strict as $a \in J''$. Hence by maximality of $J$, both $R/J'$ and $R/J''$ have a filtration as above and hence so does $R/J$. Contradiction. $\square$

Second proof. For an $R$-module $M$ we say $P(M)$ holds if there exists a filtration as in the statement of the lemma. Observe that $P$ is stable under extensions and holds for $0$. By Lemma 10.5.4 it suffices to prove $P(R/I)$ holds for every ideal $I$. If not then because $R$ is Noetherian, there is a maximal counter example $J$. By Example 10.28.7 and Proposition 10.28.8 the ideal $J$ is prime which is a contradiction. $\square$

Comment #6765 by Thomas on

In Lemma 00L0, could you write:

$aR/(J\cap aR)$ is a quotient of $R/(J+bR)$ and therefore isomorphic to $R/J'$ for some ideal $J'$ containing $J+bR,$ or even:

$aR/(J\cap aR)$ is isomorphic to $R/(J:a)$ and $J+bR \subset (J:a).$

If we start with a filtration $0=N_0\subset N_1\subset\dots \subset N_s = R/(J+bR)$ with quotients of the form $R/P$ for some prime $P,$ and then project this filtration to the quotient $aR/(J\cap aR),$ we would not necessarily get a filtration of $aR/(J\cap aR)$ with quotients of the form $R/P$ for some prime $P.$ But the text might indicate that this is the case.

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