How is primary decomposition like factoring integers?

Writing an integer as a product of prime powers corresponds, for the ideal it generates, to an intersection of primary ideals whose radicals are the primes. Primary decomposition extends this from the integers to ideals in any Noetherian ring, where literal factorization may fail.

Is a primary decomposition unique?

Not entirely. The set of associated primes and the components belonging to the minimal primes are unique, but components for embedded primes can be chosen in different ways. So the prime data is canonical while the specific components are not.

Primary Decomposition

Primary decomposition expresses an ideal in a Noetherian ring as a finite intersection of primary ideals, generalizing the factorization of integers into prime powers and revealing the associated primes.

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Definition

A primary decomposition of an ideal is an expression of it as a finite intersection of primary ideals, where an ideal is primary if a product lying in it forces one factor in it or a power of the other in it; the radicals of these components are the associated primes.

Scope

This topic covers primary ideals and their radicals, the Lasker-Noether theorem on the existence of primary decompositions in Noetherian rings, irredundant decompositions, the uniqueness of the associated primes and of the isolated primary components, and the geometric interpretation through irreducible components and embedded primes.

Core questions

What is a primary ideal, and how does it generalize a prime power?
When does an ideal admit a primary decomposition?
Which parts of a primary decomposition are uniquely determined?
How do associated and embedded primes appear geometrically?

Key theories

Lasker-Noether theorem: In a Noetherian ring every ideal is a finite intersection of primary ideals, so primary decomposition always exists, generalizing unique factorization from elements to ideals.
Uniqueness of associated primes: Although the primary components themselves are not always unique, the set of associated primes (the radicals of the components) is uniquely determined by the ideal, as are the components for the minimal associated primes.
Geometric interpretation: The minimal associated primes correspond to the irreducible components of the algebraic set defined by the ideal, while embedded primes record extra, lower-dimensional structure such as multiplicities along subvarieties.

Clinical relevance

Primary decomposition is the ideal-theoretic analogue of factorization and is foundational to algebraic geometry: it decomposes an algebraic set into irreducible components and detects embedded and multiple structure, and it organizes the associated primes of a module used throughout commutative algebra.

History

Emanuel Lasker proved primary decomposition for polynomial rings in 1905, and Emmy Noether established it abstractly for all Noetherian rings in 1921, in the paper that introduced the ascending chain condition; the result is named the Lasker-Noether theorem after them.

Key figures

Emanuel Lasker
Emmy Noether
Wolfgang Krull

Seminal works

atiyah1969
eisenbud1995
matsumura1989

Frequently asked questions

How is primary decomposition like factoring integers?: Writing an integer as a product of prime powers corresponds, for the ideal it generates, to an intersection of primary ideals whose radicals are the primes. Primary decomposition extends this from the integers to ideals in any Noetherian ring, where literal factorization may fail.
Is a primary decomposition unique?: Not entirely. The set of associated primes and the components belonging to the minimal primes are unique, but components for embedded primes can be chosen in different ways. So the prime data is canonical while the specific components are not.