Primary Decomposition

The expression of an ideal (or a submodule) in a Noetherian ring as a finite intersection of primary ideals (or primary submodules), each of which has a radical that is a prime ideal; associated primes appear as radicals of the primary components.

The expression of an ideal I in a Noetherian ring (commonly a polynomial ring) as a finite intersection I = Q1 ∩ … ∩ Qr of primary ideals Qi, where each Qi has a radical Pi that is prime; this decomposition isolates the primary components corresponding to the distinct prime-associated geometric components of V(I).

The expression of a suitably finitary substructure (typically an ideal in a Noetherian ring) as an intersection of primary substructures, where each primary component has an associated prime controlling its nilpotent behavior.