Proposition 6

The following conditions are equivalent : (i) every a-module is projective, (ii) every a-module is injective, (iii) every ideal of $A$ is an injective module, (iv) $\mathrm{dh}(A) \leq 0$, (v) $A$ is semisimple, (vi) $A$ is noetherian and every a-module is flat; (vii) every complex of a-modules is split, (viii) every exact sequence of a-modules is split.

By Prop. 4, we have (i) ⇔ (ii) ⇔ (iv); by Corollary 1 to Prop. 4, we have (vi) ⇒ (iv). By X, p. 35, Example 4, we have (i) ⇒ (vii); since (ii) ⇒ (iii) and (vii) ⇒ (viii) are trivial and since (viii) ⇒ (i) follows from II, p. 39, Prop. 4, it remains to be shown (iii) ⇒ (v) and (v) ⇒ (vi); the last assertion follows from VIII, § 5, No. 1, Props. 1 and 2; finally, if every ideal of $A$ is injective, it is a direct factor in $A$, hence (iii) ⇒ (v).