Theorem 2

— Every finitely generated module $E$ over a principal ideal domain $A$ is isomorphic to a direct sum of a finite number $m$ of cyclic modules $A/a_k$, where the $a_k$ are ideals of $A$ (some of which may be zero) such that $a_1 \subset a_2 \subset \ldots \subset a_m \neq A$, and which are uniquely determined by these conditions.

If $E$ can be generated by $q$ elements then it is isomorphic to a quotient module $L/M$, where $L = A^q$ (II. p. 218). Since $M$ has finite rank $n \leq q$ (VII, p. 15, Prop. 1), the conditions of Th. 1 (VII, p. 18) are satisfied. Then in the notation of Th. 1 $L/M$ is isomorphic to a direct sum of a complement $L'$ of $M'$ in $L$ and the torsion module $M'/M$. The module $L'$ is free of finite rank $p = q - n$, so isomorphic to $A^p$. If $r$ is the smallest index such that $A\alpha_r \neq A$, then $M'/M$ is isomorphic to the direct sum of modules $A/A\alpha_i$ for $r \leq i \leq n$. The stated conditions will then be satisfied if we take $m = p + (n - r + 1)$, $a_k = (0)$ for $1 \leq k \leq p$, and $a_j = A\alpha_n , - ,$ for $1 \leq j \leq n - r + 1$. Uniqueness follows from the Cor. of VII, p. 16.