I'm supposed to show in a part of an exercise that if we have a ring $R$ that is a principal ideal domain, then for any ideal $I$ in $R$, $R/I$ will also be a PID.
So $I=(i)$ for some $i \in R$, and $R/I$ is $\{rI |r\in R\}$, and we know:
An ideal $J$ in $R/I$ will be an ideal $J$ in $R$ containing $I$.
So if $I \subset J$ and $I=(i)$, then $J$ should be $(j)$, where $j|i$. but wherefrom do I get the necessity that $J=(j)$, so that $J$ is also principal? and there will be several $j$ that divide $i$, but should it not be only one?
And then I shall use this to show a 1-1 correspondence between ideals of $\mathbb{Z}$ and $\mathbb{N}$.
Can someone support with with the proofs?