{\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}}) \times {\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}}) \longrightarrow 
\frac{\{\, \gamma \in A\backslash\{0\} \;:\; \det(\gamma) = 0 \,\}}{({\mathbb{Z}}/N{\mathbb{Z}})^*}{\raisebox{0.4ex}{,}}$$ given by $$S\big((u:v),(x:y)\big) = 
\left(\begin{array}{@{}cc@{}} ux & uy \\ vx & vy \end{array}\right)\cdot$$ Under this correspondence, the set of proper nontrivial left $A$-ideals is in bijection with the set $$\{\, {\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}}) \times (x:y) : (x:y) \in {\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}}) \,\},$$ and the right action of $A^*/({\mathbb{Z}}/N{\mathbb{Z}})^* = {\mathrm{PGL}}_2({\mathbb{Z}}/N{\mathbb{Z}})$ on left $A$-ideals is transitive and induced by the natural (transpose) action on ${\mathbb{P}}^1({\mathbb{Z}}/N{\mathbb{Z}})$.

The nonzero matrices of determinant zero, modulo $({\mathbb{Z}}/N{\mathbb{Z}})^*$, determine a hypersurface $ad = bc$, which is the image of ${\mathbb{P}}^1 \times {\mathbb{P}}^1$ by the Segre embedding in ${\mathbb{P}}^3$ (= $(A\backslash\{0\})/({\mathbb{Z}}/N{\mathbb{Z}})^*$). It is easily verified that left and right multiplication induce the standard and transpose multiplication on the first and second factors of ${\mathbb{P}}^1 \times {\mathbb{P}}^1$, respectively, under this isomorphism, from which the result follows.

Using an explicit isomorphism ${\mathcal{O}}/N{\mathcal{O}}\cong {\mathbb{M}}_2({\mathbb{Z}}/N{\mathbb{Z}})$, by this lemma we can find $[\mu]$ in $({\mathcal{O}}/N{\mathcal{O}})^*$ such that $
({\mathcal{O}}\gamma/N{\mathcal{O}}) [\mu] = I/N{\mathcal{O}},
$ using linear algebra over ${\mathbb{Z}}/N{\mathbb{Z}}$.

In Section \[sec:ideals:ellpow:lift\] we require an input $[\mu]$ which is a unit in $Rj/N{\mathcal{O}}$. Observing that $[j]$ is a unit, we see that such units form a coset of $(R/NR)^*$: $$({\mathcal{O}}/N{\mathcal{O}})^* \cap Rj/N{\mathcal{O}}= (R/NR)^*[j].$$ We 