I would be grateful if you guide me through the following question:
Suppose a commutative ring with identity, $R$, has a unique maximal ideal, say $M$. If $M$ is principal, can we show that every ideal is finitely generated?
Well, I am considering the following facts:
$1)$ Every ideal $I$ of $R$ lies in a maximal ideal, and since we have a unique maximal ideal, then $I$ lies in $M=(m)$.
$2)$ We can also consider Nakayama's lemma: $IM$ is an ideal of $M$, and if IM=M then $M=0$. In particular we can set I=M.
$3)$ Another fact I can think of, is that it follows easily that all nonunits form an ideal in such a ring, which is $M$ itself.