Oka family

An Oka family of ideals in a commutative ring $R$ is a collection $\mathcal{F}$ containing $R$ such that, for every ideal $I$ and $a\in R$, membership of both $I+(a)$ and $(I:a)$ implies $I\in \mathcal{F}$. The requirement $R\in \mathcal{F}$ is part of the definition.

A maximal ideal outside such a family is a prime ideal. Indeed, if $ab\in I$ but $a,b\notin I$, maximality puts $I+(a)$ in the family and also puts $(I:a)$ there because it contains $b$ strictly outside $I$. The defining implication would put $I$ in the family, a contradiction. This principle converts maximal counterexamples into prime ideals.

Every ideal maximal outside an Oka family is prime. If $xy\in I$ while $x,y\notin I$, both $I+(x)$ and $(I:x)$ properly contain $I$, contradicting the Oka condition.