Complete system of prime ideals: Difference between revisions
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The collection of ''all'' maximal ideals is complete for any ring. {{proofat|[Ring equals max-localization intersection]]}}. We can imagine this intersection as being performed in the [[total quotient ring]]. | The collection of ''all'' maximal ideals is complete for any ring. {{proofat|[[Ring equals max-localization intersection]]}}. We can imagine this intersection as being performed in the [[total quotient ring]]. | ||
Latest revision as of 16:19, 12 May 2008
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Definition
Let be a commutative unital ring. A collection of prime ideals in is termed a complete system of prime ideals for if 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.