Proposition 6

Let $M$ and $N$ be projective $A$-modules of rank 1. Then $M \otimes_A N$, $\operatorname{Hom}_A(M, N)$ and the dual $M^*$ of $M$ are projective of rank 1.

This follows immediately from formulae (2), (3) and (4).

Let us now note that every finitely generated $A$-module is isomorphic to a quotient module of $L = \mathbf{A}^{(\mathbf{N})}$; we may therefore speak of the set $F(A)$ of classes of finitely generated $A$-modules with respect to isomorphism (Set Theory, Chapter I, §6, no. 9); we denote by $P(A)$ the subset of $F(A)$ consisting of the classes of projective $A$-modules of rank 1 and by $\operatorname{cl}(M)$ the image in $P(A)$ of a projective $A$-module $M$ of rank 1. It is immediate that, for two projective $A$-modules $M$,

N of rank 1, cl(M \otimes_A N) depends only on cl(M) and cl(N); as definition we set

(6)

and an internal law of composition is thus defined on $P(A)$.