Krull's principal ideal theorem

In a Noetherian ring, every prime ideal minimal over a principal ideal has height at most one. It bounds the codimension created by a single equation.

Noetherian ring: A ring in which every ascending chain of ideals stabilizes; equivalently, every ideal is finitely generated. Its finiteness condition is the essential hypothesis behind the theorem.
Local ring: A ring with exactly one maximal ideal. After localization, the prime under study becomes the unique maximal ideal.
Krull's generalized principal ideal theorem: A Noetherian-ring theorem bounding the height of a prime minimal over an ideal by its number of generators. The principal ideal theorem is its one-generator case.
Non-Noetherian ring: A ring that does not satisfy the ascending chain condition on ideals. Without Noetherianity, the principal ideal theorem can fail.
Prime ideal: A proper ideal whose quotient ring is an integral domain. The theorem bounds the height of primes minimal over the principal ideal.
Radical of an ideal: The ideal of elements some positive power of which belongs to a given ideal. Minimal primes over an ideal are unchanged when the ideal is replaced by its radical.
Codimension: A measure of the number of independent dimensions lost when passing to a subspace or closed subset. The height bound says a single equation has minimal components of codimension at most one.
Minimal prime: A prime ideal minimal under inclusion among the prime ideals containing a specified ideal. The height bound concerns minimal primes, not every prime containing the principal ideal.
Height of a prime ideal: The supremum of lengths of strictly increasing chains of prime ideals ending at a given prime. The theorem's conclusion is precisely an upper bound on this quantity.
Nakayama's lemma: A theorem that constrains finitely generated modules over local rings when a maximal ideal acts on them. Local proofs use it to turn finite generation into strong restrictions on generators.

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