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Wikipedia / Primary decomposition

In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by .

The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings. The theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components.

It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules. This contains the case for rings as a special case, considering the ring as a module over itself, so that ideals are submodules. This also generalizes the primary decomposition form of the structure theorem for finitely generated modules over a principal ideal domain, and for the special case of polynomial rings over a field, it generalizes the decomposition of an algebraic set into a finite union of (irreducible) varieties.

The first algorithm for computing primary decompositions for polynomial rings over a field of characteristic 0 was published by Noether's student . The decomposition does not hold in general for non-commutative Noetherian rings. Noether gave an example of a non-commutative Noetherian ring with a right ideal that is not an intersection of primary ideals.

Primary decomposition of an ideal

be a Noetherian commutative ring is called primary if it is a proper ideal; equivalently, every zero-divisor in the quotient is nilpotent. The radical.

has an irredundant primary decomposition into primary ideals:

  1. .

Irredundancy means:

Moreover, this decomposition is unique in the two ways:

are in general not unique (see an example below). For the existence of the decomposition, see #Primary decomposition from associated primes below.

are called the prime divisors or the primes belonging to such that

(note this practice will conflict with the usage in the module theory).

  • are the same as the minimal prime ideals and are called isolated primes.

  • The non-minimal elements, on the other hand, are called the embedded primes.

In the case of the ring of integers, the Lasker–Noether theorem is equivalent to the fundamental theorem of arithmetic, is

Similarly, in a unique factorization domain is a unit, then the primary decomposition of the principal ideal is

Examples

The examples of the section are designed for illustrating some properties of primary decompositions, which may appear as surprising or counter-intuitive. All examples are ideals in a polynomial ring over a field .

Intersection vs. product

is

Because of the generator of degree one, is not the product of two larger ideals. A similar example is given, in two indeterminates by

Primary vs. prime power

as associated prime. It is not a power of its associated prime.

Non-uniqueness and embedded prime

For every positive integer is

The associated primes are

Example: Let N=R=k[x,y] for some field k, and let M be the ideal (xy,y2). Then M has two different minimal primary decompositions M = (y) (x, y2) = (y) (x+y,y2). The minimal prime is (y) and the embedded prime is (x,y).

Non-associated prime between two associated primes

has the (non-unique) primary decomposition

is a non associated prime ideal such that

A complicated example

Unless for very simple examples, a primary decomposition may be hard to compute and may have a very complicated output. The following example has been designed to provide just such a complicated output, while, nevertheless, being accessible to hand-written computation.

Let

be two homogeneous polynomials in over a field . That is, and is searched. For computing the primary decomposition, we suppose first that 1 is a greatest common divisor of and .

This condition implies that has no primary component of height one. As is generated by two elements, this implies that it is a complete intersection (more precisely, it defines an algebraic set, which is a complete intersection), and thus all primary components have height two. Therefore, the associated primes of are exactly the primes ideals of height two that contain .

is an associated prime of .

be the homogeneous resultant in of and . As the greatest common divisor of and is a constant, the resultant is not zero, and resultant theory implies that contains all products of by a monomial in of degree This primary component contains and , and the behavior of primary decompositions under localization shows that this primary component is

such all its generating sets involve all indeterminates.

The other primary component contains . One may prove that if and are sufficiently generic (for example if the coefficients of and are distinct indeterminates), then there is only another primary component, which is a prime ideal, and is generated by , and .

Geometric interpretation

In algebraic geometry, an affine algebraic set is defined as the set of the common zeros of an ideal of a polynomial ring

An irredundant primary decomposition

of defines a decomposition of into a union of algebraic sets , which are irreducible, as not being the union of two smaller algebraic sets.

is the associated prime and Lasker–Noether theorem shows that has a unique irredundant decomposition into irreducible algebraic varieties

where the union is restricted to minimal associated primes. These minimal associated primes are the primary components of the radical of . For this reason, the primary decomposition of the radical of is sometimes called the prime decomposition of .

The components of a primary decomposition (as well as of the algebraic set decomposition) corresponding to minimal primes are said isolated, and the others are said .

For the decomposition of algebraic varieties, only the minimal primes are interesting, but in intersection theory, and, more generally in scheme theory, the complete primary decomposition has a geometric meaning.

Primary decomposition from associated primes

Nowadays, it is common to do primary decomposition of ideals and modules within the theory of associated primes. Bourbaki's influential textbook Algèbre commutative, in particular, takes this approach.

a module over it. By definition, an associated prime is a prime ideal which is the annihilator.

is nonzero.

. Directly from the definition,

, then there is a finite ascending sequence of submodules

.; i.e.,

  1. .

is a finite set is finitely generated.

and

is called -primary, the above decomposition is precisely a primary decomposition of an ideal.

(without finite generation, there can be infinitely many associated primes.)

Properties of associated primes

be a Noetherian ring. Then

Non-Noetherian case

The next theorem gives necessary and sufficient conditions for a ring to have primary decompositions for its ideals.

The proof is given at Chapter 4 of Atiyah–Macdonald as a series of exercises.

There is the following uniqueness theorem for an ideal having a primary decomposition.

..

-th symbolic power.

Additive theory of ideals

This result is the first in an area now known as the additive theory of ideals, which studies the ways of representing an ideal as the intersection of a special class of ideals. The decision on the "special class", e.g., primary ideals, is a problem in itself. In the case of non-commutative rings, the class of tertiary ideals is a useful substitute for the class of primary ideals.

Notes

References

  • , esp. section 3.3.

  • . English translation in Communications in Computer Algebra 32/3 (1998): 8–30.