Lemma 2

Let M be an A-module that is the direct sum of a family $(M_i)_{i\in I}$ of primordial submodules, and let $u$ be an endomorphism of M. Set $v= 1_M-u$ and $M_J=\bigoplus_{i\in J}M_i$ for every subset J of I. Then one of the following two properties holds:

a) There exists an index $i\in I$ such that $u$ induces an isomorphism from $M_i$ to a direct factor submodule of M.

b) For every finite subset J of I$,v$ induces an isomorphism from $M_J$ to a submodule supplementary to $M_{I-J}$.

If property b) holds, then $v$ is injective.

Suppose that property a) does not hold, and let us establish property b) by induction on the cardinal of J. There is nothing to prove if $J =\emptyset$. Therefore, assume that J is nonempty, choose an element $i$ of J, and set $J'= J-\{i\}$. By the induction hypothesis, $v$ induces an isomorphism from $M_{J'}$ to a submodule of M supplementary to $M_{I-J'}= M_{I-J}\oplus M_i$; consequently, the submodule $M''=v(M_{J'})\oplus M_{I-J}$ is supplementary to $M_i$. By Lemma 1 and the assumption on $u$, the endomorphism $v$ induces an isomorphism from $M_i$ to a submodule of M supplementary to $M''$; consequently, $v$ induces an isomorphism from $M_J= M_i\oplus M_{J'}$ to a submodule supplementary to $M_{I-J}$.

The last assertion follows from the fact that M is the union of the submodules $M_J$, where J runs through the finite subsets of I.