  The $d^2$ Pauli operators attached to a composite qudit in dimension $d$ may
be mapped to the vectors of the symplectic module $\mathcal{Z}_d^{2}$
($\mathcal{Z}_d$ the modular ring). As a result, perpendicular vectors
correspond to commuting operators, a free cyclic submodule to a maximal
commuting set, and disjoint such sets to mutually unbiased bases. For
dimensions $d=6,~10,~15,~12$, and 18, the fine structure and the incidence
between maximal commuting sets is found to reproduce the projective line over
the rings $\mathcal{Z}_{6}$, $\mathcal{Z}_{10}$, $\mathcal{Z}_{15}$,
$\mathcal{Z}_6 \times \mathbf{F}_4$ and $\mathcal{Z}_6 \times \mathcal{Z}_3$,
respectively.
