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

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

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

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

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

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

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

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

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.