Theorem 2

— Let $\mathbf{A}$ be a principal ideal domain and let $(p_\alpha)$ be a system of representatives of irreducible elements of $\mathbf{A}$. Then every nonzero element $x$ of the field of fractions of $A$ can be uniquely expressed in the form

where $u$ is an invertible element of $\mathbf{A}$, and where the $n_{\alpha}$ are integers, all but finitely many of which are zero. For $x$ to belong to $\mathbf{A}$ it is necessary and sufficient that all the $n_{\alpha}$ be positive.

We will use the theorem about decomposition as a sum of irreducible elements (VI, p. 18, Th. 2), of which the above statement is only a translation. Since $\mathscr{P}^*$ is a lattice ordered group it will be sufficient for us to show that every nonempty set of principal ideals of $\mathbf{A}$ contains a maximal element, in order to check that the hypotheses of this theorem are indeed satisfied; now this follows from the next Lemma: