= Oka family
{c}
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>.
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