##  [Krull Intersection Theorem](/krull-intersection-theorem) 

  ##  [Krull Intersection Theorem](https://algebra.quantumdictionary.io/krull-intersection-theorem-0) 

  

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- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A structural result about the I-adic topology on rings: under standard hypotheses (for example R Noetherian and I contained in the Jacobson radical, often in the local or complete Noetherian case) the intersection of all powers of a proper ideal I, ⋂_{n≥1} I^n, equals {0}. The theorem formalizes when the natural map from R to its I-adic completion is injective, i.e. when R is I-adically separated.