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
Top images from around the web for Definitions and Properties
PCK Map for Algebraic Expressions - Mathematics for Teaching View original
Is this image relevant?
9.4 – Algebraic Operations with Radical Expressions | Hunter College – MATH101 View original
Is this image relevant?
Solve Radical Equations – Intermediate Algebra but cloned this time not imported View original
Is this image relevant?
PCK Map for Algebraic Expressions - Mathematics for Teaching View original
Is this image relevant?
9.4 – Algebraic Operations with Radical Expressions | Hunter College – MATH101 View original
Is this image relevant?
1 of 3
Top images from around the web for Definitions and Properties
PCK Map for Algebraic Expressions - Mathematics for Teaching View original
Is this image relevant?
9.4 – Algebraic Operations with Radical Expressions | Hunter College – MATH101 View original
Is this image relevant?
Solve Radical Equations – Intermediate Algebra but cloned this time not imported View original
Is this image relevant?
PCK Map for Algebraic Expressions - Mathematics for Teaching View original
Is this image relevant?
9.4 – Algebraic Operations with Radical Expressions | Hunter College – MATH101 View original
Is this image relevant?
1 of 3
A proper ideal Q in a commutative ring R is called primary if for any a,b∈R with ab∈Q, either a∈Q or bn∈Q for some positive integer n
For a Q, the radical of Q, denoted by Q, is a prime ideal
This prime ideal is called the 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 Z, the ideal (pn) 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 (x2,xy) is primary with associated prime (x)
In the ring Z, the associated primes of the ideal (12) are (2) and (3)
In the ring k[x,y], the associated primes of the ideal (x2,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=Q1∩Q2∩⋯∩Qn, where each Qi 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 (): Let R be a Noetherian ring and I be an ideal of R. If I=Q1∩Q2∩⋯∩Qn=Q1′∩Q2′∩⋯∩Qm′ are two minimal primary decompositions of I, then n=m and the associated primes of Qi and Qi′ 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:
Find a decomposition of I into irreducible ideals: I=I1∩I2∩⋯∩In
For each irreducible ideal Ii, find its associated prime Pi=Ii
Group the irreducible ideals with the same associated prime, and intersect them to obtain primary ideals: Qj=⋂Pi=PjIi
Example
In the ring Z[x], consider the ideal I=(x2−1)
The primary decomposition of I is (x−1)∩(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 p such that the localization Mp 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 , as the radical is the intersection of the associated primes of the ideal's
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