System of generators for $\Bbb Z_9\times \Bbb Z_{18}$. Is $\{(4,3),(3,5)\}$ a system of generators for $\Bbb Z_9\times \Bbb Z_{18}$?
I tried to generate the elements using the straightforward method $(4,3), (8,6), (3, 9)$ ... but it takes too long.
Can I use a faster method?

Let $v_1=(4,3)$ and $v_2=(3,5)$. Then
$$
\begin{pmatrix}v_1\\v_2\end{pmatrix}
=
\begin{pmatrix}4&3\\3&5\end{pmatrix}
\begin{pmatrix}e_1\\e_2\end{pmatrix}
$$
where $e_1=(1,0)$ and $e_2=(0,1)$.
The matrix has determinant $11$, which is invertible in the ring $\Bbb Z_9\times \Bbb Z_{18}$. Therefore, $v_1$ and $v_2$ generate the same additive group as $e_1$ and $e_2$.
Explicitly,
$$
\begin{pmatrix}e_1\\e_2\end{pmatrix}
=
\begin{pmatrix}7&3\\3&2\end{pmatrix}
\begin{pmatrix}v_1\\v_2\end{pmatrix}
$$
