Primary decomposition and associated primes

A proper ideal Q is primary if ab∈Q and a∉Q imply some positive power of b lies in Q. Equivalently, every zero divisor in R/Q is nilpotent. Its radical is a prime p, and one calls Q p-primary. A prime ideal is primary, but (x²) in k[x] is a thicker primary ideal with radical (x). The quotient retains a nonzero nilpotent x, so replacing Q by its radical loses information even though V(Q) and V(p) have the same underlying closed set.

The Lasker–Noether theorem states that a proper ideal in a Noetherian ring is a finite intersection of primary ideals. One may choose the decomposition irredundant with distinct radicals. The set of those radicals is intrinsic: it is the associated-prime set of R/I. Components belonging to minimal primes over I are isolated components, while larger associated primes are embedded. Minimal primary components are uniquely determined, but embedded primary components need not be unique. The existence and uniqueness statements are theorem-level facts, not consequences of simply inspecting one convenient factorization.

For an R-module M, a prime p is associated if p is the annihilator of some nonzero element m. When R is Noetherian and M finite, Ass(M) is finite, every minimal prime of Supp(M) is associated, and the zero divisors on M form the union of its associated primes. For a finite module the support is V(Ann M), so its minimal points are algebraically accessible. An embedded associated prime is one that is not minimal in that support. These definitions distinguish a geometric component from a more localized obstruction carried by a particular element.

The qualifier associated is stronger than merely belonging to the support. A prime can have M_p nonzero without being the annihilator of one element globally. Conversely, a minimal support prime cannot be omitted from Ass in the finite Noetherian setting. Localization can isolate associated behavior, but its formulas must retain the finite-module assumptions when using these clean correspondences. The practical method in a small example is to exhibit elements and compute their full annihilator ideals, then use a primary decomposition to prove no additional associated primes were missed.

In k[x,y], the ideal (x²,xy) has radical (x), so its closed set is just the y-axis. Nevertheless the class of x in the quotient is nonzero and killed by both x and y. It gives an associated prime at the origin, larger than the generic prime (x) of that axis. This embedded point does not introduce another irreducible component of the underlying closed set. It records extra nilpotent module structure concentrated at a smaller locus inside the component. A picture showing only the axis misses it.

Associated primes are also a guide for selecting nonzerodivisors: avoid their union rather than only the minimal primes. Avoiding minimal components is insufficient when an embedded associated prime remains. This distinction will matter for depth and regular sequences, where one repeatedly needs injective multiplication maps on successive quotients. Primary decomposition is often computationally demanding, but even a partial annihilator calculation can expose a decisive embedded obstruction. Claiming a complete associated-prime list still requires an argument excluding others, such as the explicit irredundant decomposition in the worked example.