Theorem 2

— Let $G$ be a filtered group. Then the following properties are equivalent : a) $G$ is isomorphic to an ordered group of the form $\mathbf{Z}^{(I)}$. b) $G$ is lattice ordered and satisfies the following condition : (MIN) Every nonempty set of positive elements of $G$ has a minimal element. c) $G$ satisfies condition (MIN) and every irreducible element of $G$ has property (P). d) $G$ is generated by its irreducible elements, and every irreducible element of $G$ has property (P).

Let us show first that a) implies b). The group $\mathbf{Z}^{(I)}$ is lattice ordered, as the direct sum of totally ordered groups. On the other hand let $E$ be a nonempty set of positive elements of $\mathbf{Z}^{(I)}$ and let $x = \sum n_\iota e_\iota$ be an element of $E$ (where $(e_\iota)$ denotes the natural basis of $\mathbf{Z}^{(I)}$) ; there are a finite number $\prod (n_\iota + 1)$ of elements $y$ of $\mathbf{Z}^{(I)}$ such that $0 \leq y \leq x$, so the set $F$ of elements of $E$ which are less than or equal to $x$ is $a$ fortiori finite ; since it is nonempty, it contains a minimal element (Set Theory, III, p. 170, Cor. 2), which is clearly a minimal element of $E$.

It is clear that b) implies c), by Prop. 14. Let us show that c) implies d). Since $G$ is filtered, it is enough (VI, p. 4, Prop. 4) to check that the set $F$ of positive elements of $G$ which are sums of irreducible elements is equal to $G_+ - \{0\}$. If this were not true, it would follow from (MIN) that the complement of $F$ in $G_+ - \{0\}$ would have a minimal element $a$; by definition $a$ is not irreducible, so is the sum of two strictly positive elements $x$ and $y$; since $x < a$ and $y < a$, these elements belong to $F$, and so are sums of irreducible elements, and it follows that so is $a$, which is a contradiction. Finally, d) implies a) by Prop. 15.

We will apply Th. 2 to the theory of divisibility in principal ideal domains (VII, p. 4) and in unique factorisation domains (AC, VII, § 3), as well as to the study of ideals in a Dedekind ring (AC, VII, § 2).