Proposition 6

a) The mapping $E\mapsto [E]'_{\mathscr{C}}$ from $\mathscr{C}$ to $K'(\mathscr{C})$ is a weakly additive function of modules.

b) Let G be an abelian group, and let $\varphi :\mathscr{C}\rightarrow G$ be a weakly additive function of modules. There exists a unique group homomorphism $u: K'(\mathscr{C})\rightarrow$ G such that we have $\varphi (E) =u([E]'_{\mathscr{C}})$ for every module E of type $\mathscr{C}$.

c) Let E and F be modules of type $\mathscr{C}$. We have $[E]'_{\mathscr{C}}= [F]'_{\mathscr{C}}$ if and only if there exists a module M of type $\mathscr{C}$ such that $E\oplus M$ is isomorphic to $F\oplus M$.

Assertion a) is obvious. Assertion b) follows from (I, §2, No. 4, p. 19, Theorem 1) and assertion c) from (I, §2, No. 4, p. 18, Proposition 6).

Since the mapping $E\mapsto [E]_{\mathscr{C}}$ from $\mathscr{C}$ to $K(\mathscr{C})$ is a weakly additive function of modules, we can deduce a homomorphism $u: K'(\mathscr{C})\rightarrow K(\mathscr{C})$ from it. This homomorphism is surjective but is not always an isomorphism (VIII, p. 191, Remark 2).

Let $R'$ be the subgroup of $\mathbf{Z}^{(\mathscr{C})}$ generated by the elements of the form $r_{\mathscr{E}}$, where $\mathscr{E}$ is a split exact sequence of A-modules of type $\mathscr{C}$, that is, by the elements of the form cl(E$'\oplus E''$)$-$ cl(E$'$)$-$ cl(E$''$), where $E'$ and $E''$ are modules of type $\mathscr{C}$. The canonical mapping from $\mathscr{C}$ to the quotient group $\mathbf{Z}^{(\mathscr{C})}/R'$ extends to a group homomorphism $v: K'(\mathscr{C})\rightarrow \mathbf{Z}^{(\mathscr{C})}/R'$. This is an isomorphism. Indeed, the canonical mapping from $\mathscr{C}$ to $K'(\mathscr{C})$ extends to a group homomorphism from $\mathbf{Z}^{(\mathscr{C})}$ to $K'(\mathscr{C})$ whose kernel contains $R'$, and therefore to a homomorphism $v':\mathbf{Z}^{(\mathscr{C})}/R'\rightarrow K'(\mathscr{C})$ by passing to the quotient; it is clear that $v$ and $v'$ are inverse bijections.

An element of $K'(\mathscr{C})$ is called effective if it is of the form $[E]'_{\mathscr{C}}$ for an A-module E of type $\mathscr{C}$. The set of effective elements of $K'(\mathscr{C})$ is denoted by $K'(\mathscr{C})^+$.