Remark

Suppose the ring $A$ artinian. One has $\operatorname{long}_A(I) = \operatorname{long}_A(D_A(A)) = \operatorname{long}(A)$ (Theorem 2, c)). Let $M$ be a finitely generated $A$-module; it admits an injective envelope isomorphic to $I^s$, where $s$ is the dimension of the socle of $M$ (Lemma 4). Hence one has $\operatorname{long}_A(M) \leq s \operatorname{long}(A)$; for equality to hold, it is necessary and sufficient that $M$ be injective. In particular, for the $A$-module $A$ to be injective, it is necessary and sufficient that its socle be of dimension 1; thus one recovers Lemma 1 of § 3, No. 7.