Proposition 10

Let A be a commutative ring, and let $\mathscr{C}$ be an additive set of classes of A-modules satisfying the following three conditions:

(i) Every A-module of type $\mathscr{C}$ is projective $(*$or, more generally, flat$*)$.

(ii) If E and F are A-modules of type $\mathscr{C}$, then the A-module $E\otimes_AF$ is also of type $\mathscr{C}$.

(iii) The A-module A is of type $\mathscr{C}$.

Then there exists a unique ring structure on the additive group $K(\mathscr{C})$ satisfying $[E]_{\mathscr{C}}[F]_{\mathscr{C}}= [E\otimes_AF]_{\mathscr{C}}$ for every pair E, F of A-modules of type $\mathscr{C}$. The unit element of $K(\mathscr{C})$ is $[A]_{\mathscr{C}}$.