I'm looking for a reference of the following result found in Mollins:
Let $\mathcal{O}$ be an order of a quadratic field $K$ with basis $[1,\omega]$. Let $\mathfrak{a}$ be an ideal of $\mathcal{O}$. As a $\mathbb{Z}$-module, $\mathfrak{a}$ has basis $[a,b+c\omega]$ with $a,b,c\in \mathbb{Z}$ and $c|a,c|b$ and $ac| N(b+c\omega)$.
Furthermore, if $\mathfrak{a}$ is invertible in $\mathcal{O}$ and primitive (i.e., not divisible by $(n)$ for any nonunit $n\in \mathbb{Z}$) then $N(\mathfrak{a}) = a$ and $c=1$
Mollins references Cohn's A Second Course in Number Theory, but doesn't provide a page or section number and the result doesn't seem to be directly stated in Cohn. What is there is: The statement about the basis is lemma 3 in Chapter 4 of Cohn, but the divisbility statements don't seem to appear anywhere.
The statement about the norm can be deduced from Theorem 3 in Chapter 7 of Cohn, but how does one prove the statement about c?
Is there a relatively quick contained proof of the whole result and where can I find it?