##  [Krull Intersection Theorem](/krull-intersection-theorem-0) 

 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.

 

 

 

 

 

 





## Principle

Principle

Adic separation: descending powers of an ideal give an I-adic topology and, when the ring satisfies Noetherianity or completeness hypotheses, no nonzero element can lie in every power. Equivalently the only element topologically indistinguishable from zero is zero itself.

 

 

 

 

 





## Demonstration

Demonstration

Example: let R = k[[x1,…,xr]] be the formal power series ring over a field k and I = (x1,…,xr). Every nonzero power series has a term of minimal total degree, so it cannot lie in every I^n; hence ⋂ I^n = {0}. This shows I-adic separation in complete local regular examples.

 

 

 

 

## Misapplication

Misapplication

Assuming the theorem holds for arbitrary rings or for ideals not contained in the Jacobson radical. For instance, applying it to a non-Noetherian ring or to an ideal that does not define the adic topology of interest can falsely imply uniqueness of expansions or injectivity of completion maps.

 

 

 

 

 





## Consequence

Consequence

When applicable, elements and modules have unique I-adic expansions and the completion functor is faithful on R; this enables arguments that reduce questions to the adically complete case and ensures that topological limits reflect algebraic vanishing.

 

 

 

 

## Reversal

Reversal

The opposite situation occurs when ⋂ I^n is nonzero: the ring is not I-adically separated, so nonzero elements are indistinguishable from zero in the I-adic topology. Such nonseparation obstructs lifting, uniqueness of expansions, and injectivity of completion maps.

 

 

 

 

 





## Boundary

Boundary

Applies to rings and ideals meeting adic hypotheses—classically Noetherian local rings (often complete) or Noetherian rings with I in the Jacobson radical. It does not automatically hold for arbitrary rings, for ideals outside the Jacobson radical, or for topologies not generated by ideal powers.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists with Nakayama-type phenomena and with the notion of completeness: Krull intersection asserts separation (injectivity into the completion) while completeness is about surjectivity onto the completion; rings can be separated but not complete and vice versa.

 

 

 

 

 





## Synthesis

Synthesis

Krull Intersection Theorem characterizes when the I-adic topology separates points by stating that, under Noetherianity and related hypotheses, no nonzero element survives in every power of I; this underpins uniqueness in adic expansions and the injectivity of completion.