Example 4

Let $\mathscr{C}$ be the set of classes of A-modules of finite length. The mapping $\operatorname{long}_A : \mathscr{C} \to \mathbf{Z}$ that sends a class of A-modules of finite length to its length is an additive function of modules (II, §1, No. 10, p. 213, Corollary 3). The results of this subsection are a generalization of the results on modules of finite length established in II, §1, No. 10, p. 212–214.

In the remainder of this subsection, we consider an additive set $\mathscr{C}$ of A-modules and an additive mapping $\varphi$ from $\mathscr{C}$ to an abelian group $G$.

Let E and $E'$ be modules of type $\mathscr{C}$; then $E \oplus E'$ is of type $\mathscr{C}$, and there exists a split exact sequence (II, §1, No. 9, p. 210)

from this, we deduce

(1)

In particular, we have $\varphi(0) = 0$.