6
$\begingroup$

Let $R$ be a unital ring, $M$ is a finitely generated $R$-module.

My question is to prove that there exist a maximal submodule in $M$. However I have no strategy to prove that except using the idea of Zorn lemma.

Can any body help me to solve this problem?

Also, please give a counter example for the case that if $M$ is not finitely generated.

Thank for reading. I beg your pardon for my poor English

  • 3
    For an example of a non-finitely generated $R$ module with no maximal submodules, take $R=\mathbb{Z}$ and $M=\mathbb{Q}$.2012-02-19
  • 2
    Do your rings have unity?2012-02-19
  • 0
    Dear Arturo Magidin the ring is unital ring.2012-02-19
  • 1
    Dear msnaber: Its not harder to prove that any proper sub-module is contained in a maximal one. Do you see how to reduce this statement to the case $M=R$? (Here $R$ is viewed as a left $R$-module.)2012-02-19

2 Answers 2