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A primary ideal Q in a ring R is one where ab ∈ Q implies a ∈ Q or bⁿ ∈ Q for some n — the radical √Q is prime, and Q is "concentrated" at √Q. The Lasker-Noether theorem states that in a Noetherian ring, every ideal decomposes as a finite intersection of primary ideals. This is the algebraic generalization of unique prime factorization of integers to ideals in higher-dimensional rings.
Unique factorization of integers — 12 = 2² × 3 — is really a statement about ideals: (12) = (4) ∩ (3), where (4) is (2)-primary and (3) is (3)-primary. Primary decomposition generalizes this to ideals in any Noetherian ring. An ideal Q is primary if ab ∈ Q implies a ∈ Q or bⁿ ∈ Q for some positive integer n. The radical √Q (the set of elements with some power in Q) is always a prime ideal P, and we say Q is P-primary. Informally, a primary ideal is "concentrated at a single prime" — its deviation from being prime is controlled, consisting only of nilpotent elements in R/Q.
The Lasker-Noether theorem asserts that in a Noetherian ring, every ideal I can be written as a finite intersection I = Q₁ ∩ ··· ∩ Qₙ of primary ideals. The decomposition is called irredundant if no Qᵢ can be removed without changing the intersection. In an irredundant decomposition, the prime ideals Pᵢ = √Qᵢ are called the associated primes of I. The first uniqueness theorem says the set {P₁, ..., Pₙ} is uniquely determined by I (independent of the decomposition). The second uniqueness theorem says that Qᵢ is uniquely determined when Pᵢ is a minimal associated prime.
The distinction between minimal and embedded associated primes is geometrically significant. Consider the ideal I = (x², xy) in k[x,y]. Its primary decomposition is (x) ∩ (x², y) = (x) ∩ (x², xy, yn) for any n ≥ 1 — the (x,y)-primary component is not unique. The minimal prime (x) corresponds to the line x = 0; the embedded prime (x,y) corresponds to the origin, which is a "special point" on that line where extra vanishing occurs. Embedded primes detect subtle geometric features — thickened points, non-reduced structure, singularities.
Primary decomposition connects to many other parts of commutative algebra. The associated primes of an ideal determine where localization is trivial versus non-trivial. The Noetherian hypothesis is essential — non-Noetherian rings can have ideals without primary decomposition. The theory extends to modules (primary decomposition of submodules), which is the framework needed for the deeper theory of associated primes and support of modules.