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.

Principle

Decompose an ideal into components whose radicals are prime: an ideal Q is p-primary if rad(Q)=p and every zero divisor on the quotient is nilpotent. A primary decomposition represents the ideal as an intersection of such Q_i, capturing local algebraic structure around primes.

Demonstration

In the integers Z, (12) = (4) ∩ (3): (4) is 2-primary (radical (2)) and (3) is 3-primary (radical (3)). This intersection reflects the factorization 12 = 2^2·3 and isolates primary behavior at the primes 2 and 3.

Misapplication

Treating primary components as uniquely determined in full generality; while minimal primary components and their associated primes are unique up to inclusion and embedded components complicate uniqueness. Another misuse is confusing primary with prime ideals.

Consequence

Primary decomposition reveals the prime-associated local pieces of an ideal, which is essential for computing radicals, multiplicities, local properties, and for algorithmic decomposition in computational algebra under Noetherian hypotheses.

Reversal

The dual viewpoint is taking radicals of components to obtain a prime decomposition: passing from primary decomposition to its radical (an intersection of prime ideals) loses nilpotent and multiplicity information.

Boundary

Existence and finiteness of primary decompositions are guaranteed in Noetherian rings; in non-Noetherian contexts decompositions may fail to exist or be infinite. Embedded components complicate uniqueness statements and require care in interpretation.

Semantic Tension

Close to but distinct from prime decomposition: prime decomposition (as intersection of prime ideals) ignores nilpotent structure captured by primary components. Primary decomposition sits between generator-level description and radical-level prime information.

Synthesis

Primary decomposition breaks an ideal into finite primary constituents whose radicals are primes, separating nilpotent and prime-local behavior; in Noetherian rings it organizes an ideal into its associated primes and the primary pieces that encode multiplicities and local structure.