An Oka family of ideals in a commutative ring is a collection containing such that, for every ideal and , membership of both and implies . The requirement is part of the definition.
A maximal ideal outside such a family is a prime ideal. Indeed, if but , maximality puts in the family and also puts there because it contains strictly outside . The defining implication would put in the family, a contradiction. This principle converts maximal counterexamples into prime ideals.
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