Why is there such an ideal? 
I don't understand the $3$rd line in the following proof, why is there an ideal $N$ s.t. $JL=sN$. I thought ''to contain means divides'' is valid only in some special rings, for example Dedekind rings (whereas the converse is true for any commutative unital rings)


 A: Let me extract the fact you'd like to prove.
Lemma: Let $R$ be a domain, $(a)$ be a principal ideal and $I$ be an ideal with $I \subseteq (a)$. Then there exists an ideal $J$ such that $I = aJ$.
Proof: Take
$$
J = (I: (a)) = \{r \in R \mid r (a) \subseteq I\} = \{r \in R \mid ra \in I\}
$$
to be the colon ideal. We claim that $I = aJ$.
$\supseteq$: Given $aj \in aJ$, then by definition of $(I: (a))$ we have $aj \in I$.
$\subseteq$: Given $b \in I \subseteq (a)$, then $b = ar$ for some $r \in R$. Since $ar = b \in I$ then $r \in (I : (a)) = J$. Thus $b = ar \in aJ$.
Returning to your question, the above lemma shows that we can take $N = (JL : (s))$.
A: I think we can show this directly; let the set $N \subset R$ be defined as
$N= \{ n \in R \mid sn \in JL \}; \tag 0$
then it is evident that
$sN \subset JL; \tag 1$
since
$JL \subset (s), \tag 2$
every $t \in JL$ is of the form $t = sa$, $a \in R$; since such $a \in N$, this shows that
$JL \subset sN, \tag 3$
so by (1) and (3) we know that
$JL = sN. \tag 4$
We show $N \subset R$ is an ideal: suppose
$n_1, n_2 \in N; \tag 4$
then
$sn_1, sn_2 \in JL, \tag 5$
and since $JL$ is an ideal of $R$,
$s(n_1 - n_2) = sn_1 - sn_2 \in JL, \tag 6$
and so
$n_1 - n_2 \in N, \; \forall n_1, n_2 \in N; \tag 7$
likewise for
$sn = t \in JL, \; n \in N \tag 8$
we have, again since $JL$ is an ideal,
$s(nr) = (sn)r = tr \in JL, \; r \in R, \tag 9$
which shows that
$nr \in N, \; \forall n \in N, \; r \in R; \tag{10}$
combining (7) and (10) shows that $N \subset R$ is an ideal in $R$.
A: We have $(s) \supseteq IL$ for principal ideals, so $(s)$ divides $IL$. Hence there exists an ideal $M$ with $JL=(s)M=sM$. Now $JL$ is a subset of $sM$ and an ideal, hence of the form $sN$ (see quasicoherent's answer).
Reference for "to contain is to divide" with principal ideals: P. Clark, Proposition $2$, page $2$.
