Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $R$ be a commutative ring with identity, $M$ an $R$-module and $N$ is a submodule of $M$. Which of the following statements true?

  1. If every submodule of $M$ is Artinian, then $M$ is Artinian.

  2. If $M/N$ is Artinian, then $M$ is Artinian.

  3. If $N$ is Artinian, then $M/N$ is Artinian.

Clearly (1) is true. (2) is false because consider the $\mathbb{Z}$-module $\mathbb{Q}$. But (3)?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

For (3): What if you take $N = 0$ and $M = \Bbb{Z}$?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.