Krull Intersection Theorem

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.