  Qudits with local dimension $d>2$ can have unique structure and uses that
qubits ($d=2$) cannot. Qudit Pauli operators provide a very useful basis of the
space of qudit states and operators. We study the structure of the qudit Pauli
group for any, including composite, $d$ in several ways. To cover composite
values of $d$, we work with modules over commutative rings, which generalize
the notion of vector spaces over fields. For any specified set of commutation
relations, we construct a set of qudit Paulis satisfying those relations. We
also study the maximum size of sets of Paulis that mutually non-commute and
sets that non-commute in pairs. Finally, we give methods to find near minimal
generating sets of Pauli subgroups, calculate the sizes of Pauli subgroups, and
find bases of logical operators for qudit stabilizer codes. Useful tools in
this study are normal forms from linear algebra over commutative rings,
including the Smith normal form, alternating Smith normal form, and Howell
normal form of matrices. Possible applications of this work include the
construction and analysis of qudit stabilizer codes, entanglement assisted
codes, parafermion codes, and fermionic Hamiltonian simulation.
