Remark

The family $(u, (n_\alpha))$ is called the decomposition of $x$ into irreducible factors; by abuse of language, we also say that the formula (1) is the decomposition of $x$ into irreducible factors. If $x = u \prod_{\alpha} p_{\alpha}^{n_{\alpha}}$ and $y = v \prod_{\alpha} p_{\alpha}^{m_{\alpha}}$ are the decompositions of $x$ and $y$ into irreducible factors, then a necessary and sufficient condition for $x$ to divide $y$ is that $n_{\alpha} \leq m_{\alpha}$, for all $\alpha$; from this we deduce the formulae

The property expressed by Th. 2 is true for a more general class of rings than principal ideal domains; we will study them later as unique factorisation domains; and we will see that polynomial rings and formal power series rings in arbitrarily many indeterminates are unique factorisation domains (Comm. Alg., VII, § 3).