information:filtering
Primary decomposition

An ideal Q of a ring is said to be primary if Q is proper and whenever xy ∈ Q, either x ∈ Q or y ∈ Q for some positive integer n. In Z, the primary ideals are precisely the ideals of the form (p) where p is prime and e is a positive integer. Thus, a primary decomposition of (n) corresponds to representing (n) as the intersection of finitely many primary ideals.

The Lasker–Noether theorem, given here, may be seen as a certain generalization of the fundamental theorem of arithmetic:

Lasker-Noether Theorem — Let R be a commutative Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals; that is:

with Q primary for all i and Rad(Q) ≠ Rad(Q) for i ≠ j. Furthermore, if:

is decomposition of I with Rad(P) ≠ Rad(P) for i ≠ j, and both decompositions of I are irredundant (meaning that no proper subset of either { Q,..., Q } or { P,..., P } yields an intersection equal to I), t = k and (after possibly renumbering the Q) Rad(Q) = Rad(P) for all i.

For any primary decomposition of I, the set of all radicals, that is, the set {Rad(Q),..., Rad(Q)} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the assassinator of the module R / I; that is, the set of all annihilators of R / I (viewed as a module over R) that are prime.