##  [Failure of Krull Intersection](/failure-krull-intersection-0) 

 Definition

A phenomenon where the Krull intersection ∩_{n≥1} I^n of powers of an ideal I in a ring R does not equal the expected minimal set (for instance, it is nonzero when theorems predict it should be zero), indicating failure of separation in the I-adic topology or breakdown of hypotheses of intersection theorems.

 

 

 

 

 

 





## Principle

Principle

Krull intersection theorems assert that, under finiteness or completeness hypotheses (Noetherian, finitely generated modules, or separated adic topologies), the intersection of all powers of an ideal collapses to the anticipated submodule (often zero); failure occurs when those hypotheses are violated.

 

 

 

 

 





## Demonstration

Demonstration

In non-Noetherian rings, valuation rings, or infinite-variable polynomial rings, one can produce ideals I for which ⋂_{n≥1} I^n contains nonzero elements — explicit constructions exploit infinite ascending chains or ideals that are not finitely generated so the I-adic filtration is not Hausdorff.

 

 

 

 

## Misapplication

Misapplication

Using Krull intersection conclusions without checking Noetherianity, finite generation, or separatedness leads to incorrect claims about unique expansions, analytic continuation in formal schemes, or deducing nilpotence from vanishing of images in all quotients.

 

 

 

 

 





## Consequence

Consequence

When the Krull intersection behaves as expected, the I-adic topology is Hausdorff and formal completions reflect the original module faithfully; when it fails, completions lose information and power-series–type arguments about uniqueness or separation break down.

 

 

 

 

## Reversal

Reversal

The reversed perspective emphasizes rings where the intersection is large (nonzero): these rings exhibit I-adic nonseparatedness and allow nontrivial elements annihilated by arbitrarily high powers of the ideal, contrary to the separated Noetherian case.

 

 

 

 

 





## Boundary

Boundary

Applies to ideals in rings and modules and to the behavior of adic filtrations; it excludes geometric or analytic intersection analogues unless explicitly translated to I-adic language and hypotheses are assessed.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises between Krull intersection as a topological separation property and as an algebraic vanishing statement: a topologically non-Hausdorff adic topology manifests algebraically as a nontrivial Krull intersection, so the two viewpoints can suggest different remedies.

 

 

 

 

 





## Synthesis

Synthesis

Failure of Krull intersection signals that the I-adic filtration is not separating: algebraic finiteness or completeness hypotheses fail, and consequently completions and formal arguments that rely on vanishing of ⋂I^n cannot be applied without further checks.