Proposition 10

(i) Let $0 \to T_1 \to T_2 \to T_3 \to 0$ be an exact sequence of finitely generated torsion $A$-modules. Then

(ii) If there exists a pseudo-isomorphism $f : T_1 \to T_2$, then $\chi(T_1) = \chi(T_2)$.

(iii) In order that $\chi(T) = 0$, it is necessary and sufficient that $T$ be pseudo-zero.

In view of Definition 4, it suffices to consider for each $p \in P$ the values of $l_p$ for the torsion modules considered. Property (i) then follows from Chapter II, § 2, no. 4, Theorem 1 and the additivity of lengths in an exact sequence (Algebra Chapter 11, § 1, no. 10, Proposition 16) and properties (ii) and (iii) follow immediately from the definitions in no. 4.