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The $d^2$ Pauli operators attached to a composite qudit in dimension $d$ maybe mapped to the vectors of the symplectic module $\mathcal{Z}_d^{2}$($\mathcal{Z}_d$ the modular ring). As a result, perpendicular vectorscorrespond to commuting operators, a free cyclic submodule to a maximalcommuting set, and disjoint such sets to mutually unbiased bases. Fordimensions $d=6,~10,~15,~12$, and 18, the fine structure and the incidencebetween maximal commuting sets is found to reproduce the projective line overthe 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.

Qudits of composite dimension, mutually unbiased bases and projective ring geometry