Structure theorem for finitely generated modules over a PID

Let \(R\) be a principal ideal domain and \(M\) a finitely generated \(R\)-module. Then there are an integer \(r\geq0\), nonzero nonunits \(r_1 \divides r_2 \divides \cdots\divides r_m\) of \(R\), and primes \(p_1,\ldots,p_s\) of \(R\), not necessarily distinct, with exponents \(e_i\geq1\), such that \[ M \cong R^r\oplus \bigoplus_{i=1}^m R/(r_i) \qquad\text{and}\qquad M \cong R^r\oplus \bigoplus_{i=1}^s R/(p_i^{e_i}). \] The first is the invariant factor decomposition and the second the elementary divisor decomposition. The rank \(r\) is unique, the \(r_i\) are unique up to units, and the \(p_i^{e_i}\) are unique up to units and reordering.