Lemma 15.96.7. Let $A$ be a ring. Let $M$, $N_1$, $N_2$ be finite projective $A$-modules. Let $s : M \to N_1 \oplus N_2$ be a split injection. There exists a finitely generated ideal $J \subset A$ with the following property: a ring map $A \to B$ factors through $A/J$ if and only if $s \otimes \text{id}_ B$ identifies $M \otimes _ A B$ with a direct sum of submodules of $N_1 \otimes _ A B \oplus N_2 \otimes _ A B$.

**Proof.**
Choose a splitting $\pi : N_1 \oplus N_2 \to M$ of $s$. Denote $q_ i : N_1 \oplus N_2 \to N_1 \oplus N_2$ the projector onto $N_ i$. Set $p_ i = \pi \circ q_ i \circ s$. Observe that $p_1 + p_2 = \text{id}_ M$. We claim $M$ is a direct sum of submodules of $N_1 \oplus N_2$ if and only if $p_1$ and $p_2$ are orthogonal projectors. Thus $J$ is the smallest ideal of $A$ such that $p_1 \circ p_1 - p_1$, $p_2 \circ p_2 - p_2$, $p_1 \circ p_2$, and $p_2 \circ p_1$ are contained in $J \otimes _ A \text{End}_ A(M)$. Some details omitted.
$\square$

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