Invariant factors of a module over a PID

Let \(R\) be a principal ideal domain and \(M\) a finitely generated \(R\)-module. By the structure theorem, there are an integer \(r\geq 0\) and nonzero nonunits \(r_1,\ldots,r_m\in R\) with \[ M\cong R^r\oplus\bigoplus_{i=1}^m R/(r_i) \qquad\text{and}\qquad r_1 \divides r_2 \divides \cdots \divides r_m, \] and \(r_1,\ldots,r_m\) are unique up to units. The elements \(r_1,\ldots,r_m\) are the invariant factors of \(M\).