Lemma 1

Let M be an A-module, $M'$ a primordial submodule of M, and $M''$ a submodule of M supplementary to $M'$. Let $u$ be an endomorphism of M. Then $u$ or $1_M-u$ induces an isomorphism from $M'$ to a submodule of M supplementary to $M''$.

Let $p$ be the projection from M onto $M'$ with kernel $M''$, and let $v$ be the restriction of $p\circ u$ to $M'$. First, suppose that $v$ is an automorphism of $M'$. Since $v$ is injective, the restriction of $u$ to $M'$ is injective, and we have $u(M')\cap M''= 0$. Since $v$ is surjective, we have $u(M')\oplus M''= M$. Consequently, $u$ induces an isomorphism from $M'$ to a submodule supplementary to $M''$ in M. Now, suppose that $v$ is not an automorphism of $M'$. Then $1_{M'}-v$ is an automorphism of $M'$ because $M'$ is primordial. Now, $1_{M'}-v$ is the restriction of $p\circ (1_M-u)$ to $M'$. The previous reasoning proves that $1_M-u$ induces an isomorphism from $M'$ to a submodule of M supplementary to $M''$.