Morita equivalence classes for crossed product of rational rotation algebras

We study the Morita equivalence classes of crossed products of rotation algebras $A_{\theta }$, where $\theta$ is a rational number, by finite and infinite cyclic subgroups of $\mathrm{S}\mathrm{L}(2,\mathbb{Z})$. We show that for any such subgroup $F$, the crossed products $A_{\theta }⋊F$ and $A_{{\theta }^{\mathrm{\prime }}}⋊F$ are strongly Morita equivalent, where both $\theta$ and ${\theta }^{\mathrm{\prime }}$ are rational. Combined with previous results for irrational values of $\theta$, our result provides a complete classification of the crossed products $A_{\theta }⋊F$ up to Morita equivalence.