Noetherian Ring - Primary Decomposition

Primary Decomposition

In the ring of integers, an arbitrary ideal is of the form (n) for some integer n (where (n) denotes the set of all integer multiples of n). If n is non-zero, and is neither 1 nor −1, by the fundamental theorem of arithmetic, there exist primes p_i, and positive integers e_i, with . In this case, the ideal (n) may be written as the intersection of the ideals (p_ie_i); that is, . This is referred to as a primary decomposition of the ideal (n).

In general, an ideal Q of a ring is said to be primary if Q is proper and whenever, either or for some positive integer n. In, the primary ideals are precisely the ideals of the form (pe) 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.

Since the fundamental theorem of arithmetic applied to a non-zero integer n that is neither 1 nor −1 also asserts uniqueness of the representation for p_i prime and e_i positive, a primary decomposition of (n) is essentially unique.

For all of the above reasons, the following theorem, referred to as the Lasker–Noether theorem, may be seen as a certain generalization of the fundamental theorem of arithmetic:

Theorem

Let R be a 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_i primary for all i and for . Furthermore, if:

is decomposition of I with for, and both decompositions of I are irredundant (meaning that no proper subset of either or yields an intersection equal to I), and (after possibly renumbering the Q_i's) for all i.

For any primary decomposition of I, the set of all radicals, that is, the set 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.