Lemma 4

Let M be an A-module that is the direct sum of a family $(M_i)_{i\in I}$ of primordial submodules, let L be a primordial A-module, and let N be a direct factor submodule of M. Suppose that N is the direct sum of a family $(N_j)_{j\in J}$ of submodules isomorphic to L, and denote by $I_L$ the set of indices $i\in I$ such that $M_i$ is isomorphic to L. We then have

(1) Card(J) $\leqslant$ Card(I$_L$).

Let $N_0$ be a submodule of M supplementary to N. The module M is the direct sum of $N_0$ and the family $(N_j)_{j\in J}$. For every $j\in J$, denote by $p_j$ the projector of M with image $N_j$ associated with this decomposition (II, §1, No. 8, p. 209, Proposition 12). For every $i\in I$, denote by $J(i)$ the set of indices $j\in J$ such that $p_j$ induces an isomorphism from $M_i$ to $N_j$. This set is finite: indeed, if $x$ is a nonzero element of $M_i$ and K is a finite subset of J such that $x$ belongs to $N_0+\sum_{k\in K}N_k$, then we have $p_j(x) = 0$ for $j\in J$- K, so that $J(i)$ is contained in K.

Let $j\in J$. By Lemma 3, there exists an index $i\in I$ such that $p_j$ induces an isomorphism from $M_i$ to a direct factor submodule of $N_j$. Since $M_i$ is nonzero and $N_j$ is primordial and therefore indecomposable (VIII, p. 31, Proposition 4), we have $p_j(M_i) = N_j$, and $j$ belongs to $J(i)$. Since the module $M_i$ is isomorphic to $N_j$ and hence to L, the index $i$ belongs to $I_L$. This proves that J is the union of the family of finite sets $(J(i))_{i\in I_L}$. If the set J is infinite, then the set $I_L$ is infinite, and we have (Set Theory, III, §6, No. 3, p. 188, Corollary 3)

Card(J) $\leqslant \sum_{i\in I_L}$ Card(J($i$))$\leqslant$ Card(I$_L$).

Now, suppose that the set J is finite, and let us prove the lemma by induction on the cardinal of J. If J is empty, there is nothing to prove. Therefore, suppose that J is nonempty, and choose an element $j$ of J. By the above, there exists an index $i\in I_L$ such that $p_j$ induces an isomorphism from $M_i$ to $N_j$. Set $I'= I-\{i\}$ and $J'= J-\{j\}$. The module M is the direct sum of $M_i$ and the kernel of $p_j$. It is also the direct sum of $M_i$ and the submodule $M'=\oplus_{i'\in I'}M_{i'}$. Hence, there exists (II, §1, No. 9, p. 210, Corollary of Proposition 13) an isomorphism $\varphi$ from Ker $p_j= N_0\oplus_{j'\in J'}N_{j'}$ to $M'$. Set $N'=\varphi (\sum_{j'\in J'}N_{j'})$. The submodule $N'$ of $M'$ is a direct factor and is the direct sum of the family $(\varphi (N_{j'}))_{j'\in J'}$ of primordial submodules isomorphic to L. Let us apply the induction hypothesis to $M'$ and $N':$ we have Card(J$'$)$\leqslant$ Card(I$_L-\{i\}$) and therefore inequality (1).

Let us prove Theorem 1. Let $(M_i)_{i\in I}$ and $(N_j)_{j\in J}$ be two families of primordial submodules with direct sum M. Let $I_L$ (resp. $J_L$) be the set of $i\in I$ (resp. $j\in J$) such that $M_i$ (resp. $N_j$) is isomorphic to L. We have Card(J$_L$)$\leqslant$ Card(I$_L$) by Lemma 4. By interchanging the roles of I and J, we obtain the inverse inequality and thus the theorem.

The cardinal [M : L] defined in Theorem 1 is called the primordial multiplicity of L in M.