The Stacks project

Lemma 15.15.4. Let $R$ be a ring. The following are equivalent

  1. $R$ has property (P) of Lemma 15.15.2,

  2. any injective map of projective $R$-modules is universally injective,

  3. 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,

  4. if $N \subset M$ and $N$, $M$ are finite projective as $R$-modules, then $N$ is a direct summand of $M$, and

  5. any injective map $R \to R^{\oplus n}$ is a split injection.

Proof. The implication (1) $\Rightarrow $ (2) is Lemma 15.15.3. It is clear that (3) and (4) are equivalent. We have (2) $\Rightarrow $ (3), (4) by Algebra, Lemma 10.82.4. Part (5) is a special case of (4). Assume (5). Let $I = (a_1, \ldots , a_ n)$ be a proper finitely generated ideal of $R$. As $I \not= R$ we see that $R \to R^{\oplus n}$, $x \mapsto (xa_1, \ldots , xa_ n)$ is not a split injection. Hence it has a nonzero kernel and we conclude that $\text{Ann}_ R(I) \not= 0$. Thus (1) holds. $\square$

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