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.

## Comments (0)