Theorem 5

Let T be a finitely generated torsion A-module. There exist two finite families $(n_i)_{i \in I}$ and $(p_i)_{j \in I}$, where the $n_i$ are integers $\geq 1$ and the $p_i$ are prime ideals of $A$ of height 1 such that, if we write $T' = \bigoplus_{i \in I} A/p_i^{n_i}$, there exists a pseudo-isomorphism of $T$ to $T'$. Moreover, the families $(n_i)_{i \in I}$ and $(p_i)_{i \in I}$ with this property are unique to within a bijection of the indexing set and the $p_i$ contain the annihilator of $T$.

Uniqueness: Iff: $T \to T'$ is a pseudo-isomorphism and $p \in P, f_p : T, \to T'_p$ is an isomorphism. Now, $T'_p$ is the direct sum of the $A_p/p^{n_i}A_p$, the sum being over the indices $i$ such that $p_i = p$; the $p^{n_i}A_p$ are therefore the elementary divisors of the torsion $A$-module $T$, (Algebra, Chapter VII, § 4, no. 7); their uniqueness has been proved in Algebra, Chapter VII, § 4, no. 7, Proposition 7.

Existence: We may confine our attention to the case where $T \neq 0$. Let $a$ be the annihilator (non-zero and distinct from $A$) of $T$, $p_i$ ($1 \leq i \leq k$) the prime ideals of $A$ of height 1 containing $a$ (which are finite in number ($§ 1$, no. 6, Theorem 4)) and $S = \bigcap_i (A - p_i)$. The semi-local ring $A' = S^{-1}A$ is a principal ideal domain (Lemma 2) and has maximal ideals the $m_i = p_iA'$; as $S^{-1}T$ is a finitely generated torsion $A$-module, it is isomorphic to a finite direct sum $\bigoplus_{j \in I} A'/m_{\phi(j)}^{n_j}$, where $\phi$ is a mapping of a finite set $I$ to $[1, k]$ (Algebra, Chapter VII, § 4, no. 7, Proposition 7); as $A'/m_{\phi(j)}^{n_j}$ is isomorphic to $S^{-1}(A/p_{\phi(j)}^{n_j})$ (Chapter II, § 2, no. 4), we have obtained a torsion $A$-module $T'$ of the desired type and an isomorphism $f_0$ of $S^{-1}T$ onto $S^{-1}T'$. As $\operatorname{Hom}_{S^{-1}A}(S^{-1}T, S^{-1}T')$ is equal to $S^{-1} \operatorname{Hom}_A(T, T')$ (Chapter 11, § 2, no. 7, Proposition 19), there exist $s \in S$ and a homomorphism $f : T \to T'$ such that $f_0 = s^{-1}f$. It remains to show that $f$ is a pseudo-isomorphism: now, if $p = 0$ or if $p \in P$ is distinct from the $p_i$, $T_p = T'_p = 0$ (Chapter II, § 4, no. 4, Proposition 17); if on the contrary $p$ is one of the $p_i$ ($1 \leq i \leq k$), $s$ is invertible in $A_p$, and, as $f_{p_i} = s(f_0)_{p_i}$ and $(f_0)_{p_i}$ is an isomorphism of $T_{p_i} = (S^{-1}T)_{m_i}$ onto $T'_{p_i} = (S^{-1}T')_{m_i}$, so is $f_{p_i}$.