Complete system of prime ideals

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Definition

Let $R$ be a commutative unital ring. A collection of prime ideals in $R$ is termed a complete system of prime ideals for $R$ if $R$ is the intersection of its localizations at each of these prime ideals.

Facts

The collection of all maximal ideals is complete for any ring. For full proof, refer: Ring equals max-localization intersection. We can imagine this intersection as being performed in the total quotient ring.