going-up theorem

Picture the prime ideals of a ring stacked into towers, each tower a chain of nested primes — this is the skeleton that measures a ring's dimension. The going-up theorem says that when you sit a ring on top of a base via an integral extension, you can always lift a tower upstairs: any chain of primes downstairs can be matched by a chain upstairs lying over it, climbing in step. Integral extensions never lose vertical room.

Let R ⊆ S be an integral extension of commutative rings. Going-up states: given a chain of primes p_1 ⊆ p_2 ⊆ ... ⊆ p_n in R, and a prime q_1 of S lying over p_1 (meaning q_1 ∩ R = p_1), there exists a chain q_1 ⊆ q_2 ⊆ ... ⊆ q_n in S with each q_i lying over p_i. The companion lying-over theorem guarantees the starting q_1 exists at all, and incomparability says no two distinct primes of S over the same prime of R are comparable.

A central consequence: an integral extension preserves Krull dimension, dim S = dim R. Going-up is one of two cousins; the going-down theorem lifts chains the other direction but needs an extra hypothesis (e.g. R normal and S a domain), and can genuinely fail without it. The classic application is in algebraic number theory and in proving Noether normalization gives the right dimension count.