1

Let $N$ be a submodule of $R$-module of $M$. Prove that if $N$ and $M/N$ satisfy DCC then so $M$ does.

Thanks in advance.

egreg
  • 238,574
Truong
  • 4,162
  • What are your thoughts about the problem? – egreg Nov 24 '13 at 14:55
  • I think this theorem can usefull:

    "Let $N$ be a submodule of $R$-module $M$. Then there has a corresponding $1-1$ between the set of submodules of $M$ containg $N$ and the set of submodules of module $M/N$".

     – Truong Nov 24 '13 at 15:18

1 Answers1

1

Hint: If $L_0\supseteq L_1\supseteq\dots\supseteq L_n\supseteq\dotsb$ is an a descending chain of submodules in $M$, then also

$$L_0+N\supseteq L_1+N\supseteq\dots\supseteq L_n+N\supseteq\dotsb$$

is a descending chain. By the correspondence theorem and the fact that $M/N$ satisfies the descending chain condition, then …

Moreover also

$$L_0\cap N\supseteq L_1\cap N\supseteq\dots\supseteq L_n\cap N\supseteq\dotsb$$

is a descending chain. Therefore …

More hints. There is $n$ such that $L_m+N=L_n+N$ and $L_m\cap N=L_n\cap N$, for all $m\ge n$, because the two descending chains are stationary. The first one is because it defines a descending chain in $M/N$ which is stationary by hypothesis.

Since the chains are only two, we can choose a point from which they are both stationary.

Can we say that $L_m=L_n$ for all $m\ge n$?

egreg
  • 238,574