Invariant factors and elementary divisors of a finitely generated module over a PID

Let $R$ be a principal ideal domain and $M$ a finitely generated $R$-module. In a decomposition

$M\cong R^r\oplus R\mathrm{/}(a_1)\oplus \cdots \oplus R\mathrm{/}(a_t),a_1\mid a_2\mid \cdots \mid a_t,$

where every $a_i$ is a nonzero nonunit, the associate classes of $a_1,\dots ,a_t$ are the invariant factors of $M$. Unit factors are omitted.

After each $a_i$ is factored into powers of irreducibles and the coprime cyclic quotients are split, the resulting prime powers $p^e$, counted with multiplicity and up to associates, are the elementary divisors of $M$. The summands belonging to one associate class of $p$ form the $p$-primary component (The $p$-primary component of a module over a domain). The integer $r$ is separate data and is not an elementary divisor.