primary decomposition

Integers factor uniquely into prime powers; ideals would love to do the same. Primary decomposition is the ideal-theoretic analogue of writing 12 = 4 · 3 = 2^2 · 3 — it breaks an ideal into 'prime-power-like' pieces called primary ideals, each focused on a single prime. It is how a complicated subvariety gets resolved into its irreducible components together with the embedded fuzz around them.

A primary ideal Q is one where every zero-divisor in R/Q is nilpotent; equivalently if ab lies in Q then a lies in Q or some power of b lies in Q. The radical of a primary ideal is prime, called its associated prime, and Q is then 'P-primary'. The Lasker–Noether theorem states that in a Noetherian ring every ideal I is a finite intersection I = Q_1 ∩ ... ∩ Q_r of primary ideals, and one may take this decomposition to be minimal (irredundant, with distinct associated primes).

Uniqueness is subtle and only partial. The set of associated primes is uniquely determined by I, and the primary components belonging to minimal (isolated) primes are unique. But components belonging to embedded primes — primes strictly containing another associated prime — are not unique. The classic example (x^2, xy) = (x) ∩ (x^2, xy, y^2) = (x) ∩ (x^2, y) shows two different minimal decompositions sharing the embedded prime (x, y).