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 Lasker–Noether 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.
Let "R" be a Noetherian commutative ring. An ideal "I" of "R" is called "primary" if it is a proper ideal and for each pair of elements "x" and "y" in "R" such that "xy" is in "I", either "x" or some power of "y" is in "I"; equivalently, every zero-divisor in the quotient "R"/"I" is nilpotent. The radical of a primary ideal "Q" is a prime ideal and "Q" is said to be formula_1-primary for formula_2.
Let "I" be an ideal in "R". Then "I" has an irredundant primary decomposition into primary ideals:
Irredundancy means:
Moreover, this decomposition is unique in the two ways:
Primary ideals which correspond to non-minimal prime ideals over "I" are in general not unique (see an example below). For the existence of the decomposition, see #Primary decomposition from associated primes below.
The elements of formula_7 are called the prime divisors of "I" or the primes belonging to "I". In the language of module theory, as discussed below, the set formula_7 is also the set of associated primes of the formula_16-module formula_17. Explicitly, that means that there exist elements formula_18 in "R" such that
By a way of shortcut, some authors call an associated prime of formula_17 simply an associated prime of "I" (note this practice will conflict with the usage in the module theory).
In the case of the ring of integers formula_22, the Lasker–Noether theorem is equivalent to the fundamental theorem of arithmetic. If an integer "n" has prime factorization formula_23, then the primary decomposition of the ideal formula_24 generated by in formula_22, is
Similarly, in a unique factorization domain, if an element has a prime factorization formula_27 where is a unit, then the primary decomposition of the principal ideal generated by 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.
The primary decomposition in formula_29 of the ideal formula_30 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.
In formula_33, the ideal formula_34 is a primary ideal that has formula_35 as associated prime. It is not a power of its associated prime.
Non-uniqueness and embedded prime.
For every positive integer , a primary decomposition in formula_33 of the ideal formula_37 is
The associated primes are 
Example: Let "N" = "R" = "k"["x", "y"] for some field "k", and let "M" be the ideal ("xy", "y"2). Then "M" has two different minimal primary decompositions
"M" = ("y") ∩ ("x", "y"2) = ("y") ∩ ("x" + "y", "y"2).
The minimal prime is ("y") and the embedded prime is ("x", "y").
Non-associated prime between two associated primes.
In formula_40 the ideal formula_41 has the (non-unique) primary decomposition
The associated prime ideals are formula_43 and formula_44 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 for providing such a complicated output, and, nevertheless, being accessible to hand-written computation.
Let
be two homogeneous polynomials in , whose coefficients formula_47 are polynomials in other indeterminates formula_48 over a field . That is, and belong to formula_49 and it is in this ring that a primary decomposition of the ideal formula_50 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 .
It follows that formula_51 is an associated prime of .
Let formula_52 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 . As formula_53 all these monomials belong to the primary component contained in formula_54 This primary component contains and , and the behavior of primary decompositions under localization shows that this primary component is 
In short, we have a primary component, with the very simple associated prime formula_56 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 formula_57
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.
If formula_59 is the associated prime of formula_4, then formula_61 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.
Let "R" be a ring and "M" a module over it. By definition, an associated prime is a prime ideal appearing in the set formula_63 = the set of annihilators of nonzero elements of "M". Equivalently, a prime ideal formula_1 is an associated prime of "M" if there is an injection of an "R"-module formula_65.
A maximal element of the set of annihilators of nonzero elements of "M" can be shown to be a prime ideal and thus, when "R" is a Noetherian ring, "M" is nonzero if and only if there exists an associated prime of "M".
The set of associated primes of "M" is denoted by formula_66 or formula_67. Directly from the definition,
If "M" is a finitely generated module over "R", then there is a finite ascending sequence of submodules
such that each quotient "M""i" /"M""i−1" is isomorphic to formula_77 for some prime ideals formula_78, each of which is necessarily in the support of "M". Moreover every associated prime of "M" occurs among the set of primes formula_78; i.e.,
(In general, these inclusions are not the equalities.) In particular, formula_67 is a finite set when "M" is finitely generated.
Let formula_82 be a finitely generated module over a Noetherian ring "R" and "N" a submodule of "M". Given formula_83, the set of associated primes of formula_84, there exist submodules formula_85 such that formula_86 and
A submodule "N" of "M" is called "formula_1-primary" if formula_89. A submodule of the "R"-module "R" is formula_1-primary as a submodule if and only if it is a formula_1-primary ideal; thus, when formula_92, the above decomposition is precisely a primary decomposition of an ideal.
Taking formula_93, the above decomposition says the set of associated primes of a finitely generated module "M" is the same as formula_94 when formula_95 (without finite generation, there can be infinitely many associated primes.)
Properties of associated primes.
Let formula_16 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.
Now, for any commutative ring "R", an ideal "I" and a minimal prime "P" over "I", the pre-image of "I" "R""P" under the localization map is the smallest "P"-primary ideal containing "I". Thus, in the setting of preceding theorem, the primary ideal "Q" corresponding to a minimal prime "P" is also the smallest "P"-primary ideal containing "I" and is called the "P"-primary component of "I".
For example, if the power "P""n" of a prime "P" has a primary decomposition, then its "P"-primary component is the "n"-th symbolic power of "P".
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.