The Stacks project

Example 10.28.5. Let us show that the family of principal ideals of a ring $R$ is an Oka family. Indeed, suppose $I \subset R$ is an ideal, $a \in R$, and $(I, a)$ and $(I : a)$ are principal. Note that $(I : a) = (I : (I, a))$. Setting $J = (I, a)$, we find that $J$ is principal and $(I : J)$ is too. By Lemma 10.28.1 we have $I = J (I : J)$. Thus we find in our situation that since $J = (I, a)$ and $(I : J)$ are principal, $I$ is principal.