Lemma 15.15.4. Let $R$ be a ring. The following are equivalent
$R$ has property (P) of Lemma 15.15.2,
any injective map of projective $R$-modules is universally injective,
if $u : N \to M$ is injective and $N$, $M$ are finite projective $R$-modules then $\mathop{\mathrm{Coker}}(u)$ is a finite projective $R$-module,
if $N \subset M$ and $N$, $M$ are finite projective as $R$-modules, then $N$ is a direct summand of $M$, and
any injective map $R \to R^{\oplus n}$ is a split injection.
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