Definition 2

Let $\varphi$ be a mapping from $\mathscr{C}$ to an abelian group G (written additively); set $\varphi (E) =\varphi$(cl(E)) for every A-module E of type $\mathscr{C}$. We say that $\varphi$ is an additive function of modules (resp. a weakly additive function of modules) if we have $\varphi (E) =\varphi (E') +\varphi (E'')$ for every exact sequence (resp. for every split exact sequence)

of modules of type $\mathscr{C}$.