2.4 Primary decomposition and associated primes

Primary decomposition breaks down ideals into simpler parts, helping us understand their structure. It's like factoring numbers, but for ideals in rings. This concept is crucial for grasping how ideals behave in commutative algebra.

Associated primes are the key players in primary decomposition. They're the "prime factors" of an ideal, giving us insight into its properties and behavior. Understanding these primes is essential for deeper algebraic analysis.

Primary Ideals and Associated Primes

Definitions and Properties

• A proper ideal $Q$ in a commutative ring $R$ is called primary if for any $a,b \in R$ with $ab \in Q$, either $a \in Q$ or $b^n \in Q$ for some positive integer $n$
• For a primary ideal $Q$, the radical of $Q$, denoted by $\sqrt{Q}$, is a prime ideal
  • This prime ideal is called the associated prime of $Q$
• The associated primes of an ideal $I$ are the radicals of the primary ideals in its minimal primary decomposition

Examples

• In the ring $\mathbb{Z}$, the ideal $(p^n)$ is primary for any prime number $p$ and positive integer $n$
  • The associated prime is $(p)$
• In the ring $k[x,y]$ (where $k$ is a field), the ideal $(x^2, xy)$ is primary with associated prime $(x)$
• In the ring $\mathbb{Z}$, the associated primes of the ideal $(12)$ are $(2)$ and $(3)$
• In the ring $k[x,y]$, the associated primes of the ideal $(x^2, xy)$ are $(x)$ and $(y)$

Primary Decomposition in Noetherian Rings

Existence Theorem and Proof

• Theorem (Existence of Primary Decomposition): Let $R$ be a Noetherian ring and $I$ be an ideal of $R$. Then $I$ can be expressed as a finite intersection of primary ideals, i.e., $I = Q_1 \cap Q_2 \cap \cdots \cap Q_n$, where each $Q_i$ is a primary ideal
• Proof (Existence): The proof relies on the fact that $R$ is Noetherian, which means that every ascending chain of ideals stabilizes
  • The key steps involve induction on the number of generators of the ideal and the use of the Noetherian property to ensure the process terminates

Uniqueness Theorem and Proof

• Theorem (Uniqueness of Primary Decomposition): Let $R$ be a Noetherian ring and $I$ be an ideal of $R$. If $I = Q_1 \cap Q_2 \cap \cdots \cap Q_n = Q'_1 \cap Q'_2 \cap \cdots \cap Q'_m$ are two minimal primary decompositions of $I$, then $n = m$ and the associated primes of $Q_i$ and $Q'_i$ are the same (up to permutation)
• Proof (Uniqueness): The proof involves showing that the associated primes in both decompositions must be the same, using the properties of primary ideals and their radicals

Computing Primary Decomposition

Algorithm

• To compute the primary decomposition of an ideal $I$ in a Noetherian ring $R$:
  1. Find a decomposition of $I$ into irreducible ideals: $I = I_1 \cap I_2 \cap \cdots \cap I_n$
  2. For each irreducible ideal $I_i$, find its associated prime $P_i = \sqrt{I_i}$
  3. Group the irreducible ideals with the same associated prime, and intersect them to obtain primary ideals: $Q_j = \bigcap_{P_i = P_j} I_i$

Example

• In the ring $\mathbb{Z}[x]$, consider the ideal $I = (x^2 - 1)$
  • The primary decomposition of $I$ is $(x-1) \cap (x+1)$
  • The associated primes are $(x-1)$ and $(x+1)$

Applications of Primary Decomposition

Structure of Ideals and Modules

• Primary decomposition can be used to determine the dimension of a ring or module, as the dimension is related to the associated primes
• It can also be used to study the support of a module $M$, which is the set of prime ideals $\mathfrak{p}$ such that the localization $M_{\mathfrak{p}}$ is non-zero
• Primary decomposition is a key tool in understanding the structure of ideals and modules over Noetherian rings, as it allows for the study of their minimal components and associated primes

Algebraic Geometry and Other Applications

• Primary decomposition has applications in algebraic geometry, where it is used to study the structure of algebraic varieties and their singularities
• It can also be used to compute the radical of an ideal, as the radical is the intersection of the associated primes of the ideal's primary components
• Primary decomposition is a powerful tool in commutative algebra with wide-ranging applications in various areas of mathematics, including algebraic geometry, number theory, and combinatorics