In this paper, we determine the number of solutions of linear congruences, as well as its generalization to principal ideal domains with a certain finiteness condition, by using the Smith normal form of the involved matrix. An explicit formula, depending only on the Smith normal form, as well as upper bounds are given. This can be used in analytic number theory to bound the value of certain sums appearing in circle methods.

linear congruences, Smith normal form, principal ideal domain, factor rings.