Given a maximal ideal $M$ in a non-Noetherian commutative ring $R$, I'm trying to determine whether or not there can exist infinite strictly ascending chains of ideals of $R$ contained in $M$. I know that since $R$ is not Noetherian:
- There exists an infinite strictly ascending chain of ideals $I_1\subset I_2\subset \dotsb$ contained in $R$, and
- There exists an ideal $I$ of $R$ that is not finitely generated.
First I wanted to try supposing that there exists an infinite strictly ascending chain of ideals of $R$ contained in $M$ and derive a contradiction. Somehow I should use the facts that $M$ is maximal and that $R$ is not Noetherian. The maximality of $M$ tells me that there are no proper ideals of $R$ properly containing $M$, but I don't see how it can tell me anything about ideals of $R$ contained inside $M$. Are there any suggestions to see how these facts fit together?