8.2 Primary decomposition of ideals

Primary decomposition is a powerful tool in commutative algebra, breaking down ideals into simpler parts. It expresses an ideal as an intersection of primary ideals, revealing its structure and associated prime ideals.

This technique has wide-ranging applications, from computing radicals to analyzing algebraic sets. Understanding primary decomposition is crucial for tackling complex problems in ring theory and algebraic geometry.

Primary Decomposition Fundamentals

Definition of primary decomposition

• Primary decomposition of ideal $I$ in ring $R$ expresses $I$ as finite intersection of primary ideals $I = Q_1 \cap Q_2 \cap ... \cap Q_n$
• Primary ideal $Q$ satisfies condition $ab \in Q$ implies $a \in Q$ or $b^n \in Q$ for some positive integer $n$ (generalizes prime ideal concept)
• Associated prime ideals $P_i = \sqrt{Q_i}$ for each $Q_i$ in decomposition reveal ideal structure
• Non-redundancy ensures no $Q_i$ contains intersection of all other $Q_j$, simplifying decomposition

Computation of primary decompositions

• Monomial ideals utilize lcm of generators, factoring each into prime powers and combining factors with same variables ($(x^2y, xy^3) = (x^2, y) \cap (x, y^3)$)
• Principal ideal domains (PIDs) factor generator into prime powers, each corresponding to primary component ($\langle 12 \rangle = \langle 2^2 \rangle \cap \langle 3 \rangle$ in $\mathbb{Z}$)
• Computation steps:
  1. Identify ideal generators
  2. Factor generators (PIDs) or separate variables (monomials)
  3. Form primary ideals from factors or variable groups
  4. Intersect primary ideals for final decomposition

Advanced Concepts and Applications

Existence and uniqueness in Noetherian rings

• Existence proof uses Noetherian induction on ideal $I$:
  1. Base case: $I$ primary
  2. Inductive step: For non-primary $I$, find $ab \in I$ with $a \notin I$ and $b^n \notin I$
  3. Form ideals $I : (a)$ and $I + (b^n)$
  4. Apply induction to smaller ideals
• Uniqueness of minimal primary decomposition $I = Q_1 \cap Q_2 \cap ... \cap Q_n$:
  • Associated primes $P_i = \sqrt{Q_i}$ unique
  • Primary components $Q_i$ for minimal primes unique
• Minimal decomposition eliminates redundant components and ensures distinct associated primes

Applications of primary decomposition

• Determine associated primes of ideal (key to understanding ideal structure)
• Find minimal primes containing ideal (geometric significance in algebraic geometry)
• Compute radical $\sqrt{I} = \sqrt{Q_1} \cap \sqrt{Q_2} \cap ... \cap \sqrt{Q_n}$ (relates to zero-set of polynomials)
• Analyze quotient ring dimension $\dim(R/I) = \max\{\dim(R/P_i)\}$ (important in algebraic geometry)
• Study embedded primes and their effects (non-minimal associated primes)
• Determine ring depth using associated primes of $(0)$ (homological property)
• Identify irreducible components of algebraic set (correspond to minimal primes)
• Solve polynomial equation systems by breaking down using primary decomposition (simplifies complex systems)