Proposition 3

Let $a, a', b, b'$ be elements of $I(A)$. The relations $a > a'$ and $b > b'$ imply $ab > a'b'$.

We may restrict our attention to the case where $b = b'$. Then let $Ax$ be a fractional principal ideal containing $a'b$; for every non-zero element $y$ of $b$, $Ax \supset a'y$ and hence $Axy^{-1} \supset a'$, whence $Axy^{-1} \supset a$ and $Ax \supset ay$. Varying $y$, it is seen that $Ax \supset ab$, whence $ab > a'b$.

It follows from Proposition 3 that multiplication on $I(A)$ defines, by passing to the quotient, a law of composition on $D(A)$ which is obviously associative and commutative. It is written additively so that we may write:

for $a, b$ in $I(A)$. Clearly $\text{div}(1)$ is an identity element for this addition; this element is denoted by 0. Proposition 3 proves further that the order structure on $D(A)$ is compatible with this addition (Algebra, Chapter VI, § 1, no. 1) and, more precisely (no. 1, Proposition 2 (ii)):

For a fractional ideal $a \neq 0$ to be such that $\text{div } a \geq 0$ in $D(A)$, it is necessary and sufficient that $a \subset A$ (in other words, that $a$ be an integral ideal of $A$).

For two elements $x, y$ of $K^*$, the relation $\text{div}(x) = \text{div}(y)$ is equivalent to $Ax = Ay$; the set of principal divisors of $A$ with the order relation and the monoid law induced by that on $D(A)$ is an ordered group canonically isomorphic to the multiplicative group of fractional principal ideals ordered by the opposite order relation to inclusion (Algebra, Chapter VI, § 1, no. 5). The relation $S$ between two elements $P, Q$ of $D(A)$:

"there exists $x \in K^*$ such that $P = Q + \text{div}(x)$"

is therefore an equivalence relation since the relation $P = Q + \text{div}(x)$ is equivalent to $Q = P + \text{div}(x^{-1})$; if $P$ and $Q$ are congruent modulo $S$, they are called equivalent divisors of $A$. Clearly moreover the relation $S$ is compatible with the law of the monoid $D(A)$ and the latter therefore defines, by taking quotients, a monoid structure on $D(A)/S$; this monoid is called the divisor class monoid of A.