Corollary 5

Let M be a semiprimordial module that is the direct sum of a finite family $(M_i)_{i\in I}$ of primordial submodules. For any subset J of I, write $M_J=\bigoplus_{i\in J}M_i$. Let N be a direct factor submodule of M.

a) There exists a subset J of I such that $M_J$ is a submodule supplementary to N.

b) Let J be a subset of I. If $M_J$ is supplementary to N, then the module N is isomorphic to $M_{I-J}$ and is semiprimordial.

We denote by K the set of indices $i\in I$ such that $N\cap M_i= 0$; let us use induction on the cardinal of K. The corollary is clear if M = N. Suppose $M\not= N$. Let $p$ be a projector of M with kernel N. Denote its image by P. It is nonzero, and by Lemma 3, there exists a $j\in I$ such that $p$ induces an isomorphism from $M_j$ to a direct factor submodule of P. We have $N\cap M_j= 0$. Set $N'= N\oplus M_j$. We have $N'= N\oplus p(M_j)$. A submodule supplementary to $p(M_j)$ in P is also supplementary to $N'$ in M, so that $N'$ is a direct factor submodule of M. The set of indices $i\in I$ such that $N'\cap M_i= 0$ is contained in K $-\{j\}$. By the induction hypothesis, there exists a subset $J'$ of I such that $M_{J'}$ is a submodule supplementary to $N'$ in M. Set $J = J'\cup \{j\}$. Then $M_J$ is a submodule supplementary to N in M.

Let J be a subset of I such that $M_J$ is a submodule supplementary to M in N. Since $M_J$ is also supplementary to $M_{I-J}$, the modules N and $M_{I-J}$ are isomorphic and N is semiprimordial.