One of the most famous forms of Nakayama's lemma says:
Let $I$ be an ideal in $R$ and $M$ a finitely-generated $R$ module.
If $IM = M$, then there exists an $r \in R$ with $r ≡ 1 \pmod I$, such that $rM = 0$.
Someone knows counterexamples to the assert IN THIS FORM if $M$ is not finitely generated?
To be more precise, I'm looking to an $R$-molude $M$ such than doesn't exists an element $r \in R$ with the properties above (and, possibly, an explicit proof of this "non existence").
Thank you.
Best Answer
Let $M = \mathbb{Q}$ thought of as a $\mathbb{Z}$-module and let $I = (2)$. Then $IM = M$, and odd numbers don't kill $\mathbb{Q}$.