associated prime

The associated primes of a module are the prime ideals 'visibly attached' to it — the primes that actually show up as the precise set of things killing some single element. If a module is a geometric object, its associated primes are the irreducible components it sits over, together with the extra embedded primes where it has hidden, lower-dimensional fuzz. They are the intrinsic skeleton that primary decomposition reconstructs.

Let M be a module over a commutative ring R. A prime ideal p is an associated prime of M if p is the annihilator of some single element x of M, i.e. p = Ann(x) = { r in R : rx = 0 } for some x. The set of these is written Ass(M). Equivalently p is associated iff R/p embeds as a submodule of M. For a Noetherian ring and finitely generated M, Ass(M) is finite and nonempty whenever M is nonzero.

Two structural facts make them powerful. First, the set of zero-divisors on M is exactly the union of the associated primes — so the associated primes detect precisely where multiplication fails to be injective. Second, applied to M = R/I, the associated primes Ass(R/I) are exactly the primes appearing in a minimal primary decomposition of I, recovering both the minimal (isolated) primes, those minimal over I, and the embedded ones, which strictly contain another associated prime.