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.
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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