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. 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 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.
Definitions
Write R for a commutative ring, and M and N for modules over it.- A zero divisor of a module M is an element x of R such that xm = 0 for some non-zero m in M.
- An element x of R is called nilpotent in M if xnM = 0 for some positive integer n.
- A module M is called coprimary if every zero divisor of M is nilpotent in M. For example, groups of prime power order and free abelian groups are coprimary modules over the ring of integers.
- A submodule M of a module N is called a primary submodule if N/M is coprimary.
- An ideal I is called primary if it is a primary submodule of R. This is equivalent to saying that if ab is in I then either a is in I or bn is in I for some n, and to the condition that every zero-divisor of the ring R/I is nilpotent.
- A submodule M of a module N is called irreducible if it is not an intersection of two strictly larger submodules.
- An associated prime of a module M is a prime ideal that is the annihilator of some nonzero element of M.
Statement
An equivalent statement is: every finitely generated module over a Noetherian ring is contained in a finite product of coprimary modules.
The Lasker–Noether theorem follows immediately from the following three facts:
- Any submodule of a finitely generated module over a Noetherian ring is an intersection of a finite number of irreducible submodules.
- If M is an irreducible submodule of a finitely generated module N over a Noetherian ring then N/M has only one associated prime ideal.
- A finitely generated module over a Noetherian ring is coprimary if and only if it has at most one associated prime.
Primary decomposition of ideals
Let R be a Noetherian commutative ring, and I an ideal in R. Then I has an irredundant primary decomposition into primary ideals.Irredundancy means:
- Removing any of the changes the intersection, i.e. for all i we have
- The associated prime ideals are distinct.
In the case of the ring of integers, the Lasker–Noether theorem is equivalent to the fundamental theorem of arithmetic. If an integer n has prime factorization, then the primary decomposition of the ideal generated by in, is
Similarly, in a unique factorization domain, if an element has a prime factorization 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 of the ideal isBecause 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, the ideal is a primary ideal that has as associated prime. It is not a power of its associated prime.Non-uniqueness and non-minimal associated prime
For every positive integer, a primary decomposition in of the ideal isThe associated primes are
Non-associated prime between two associated primes
In the ideal has the primary decompositionThe associated prime ideals are and 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 are polynomials in other indeterminates over a field. That is, and belong to and it is in this ring that a primary decomposition of the ideal 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, 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 is an associated prime of.
Let 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 all these monomials belong to the primary component contained in 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 such all its generating sets involve all indeterminates.
The other primary component contains. One may prove that if and are sufficiently generic, 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 ringAn 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 is the associated prime of, then 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 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.
Proof
Nowadays, it is common to do primary decomposition within the theory of associated primes. The proof below is in the spirit of this approach.Let M be a finitely generated module over a Noetherian ring R and N a submodule. To show N admits a primary decomposition, by replacing M by, it is enough to show that when. Now,
where are primary submodules of M. In other words, 0 has a primary decomposition if, for each associated prime P of M, there is a primary submodule Q such that. Now, consider the set . The set has a maximal element Q since M is a Noetherian module. If Q is not P-primary, say, is associated with, then for some submodule Q
Remark: The same proof shows that if R, M, N are all graded, then in the decomposition may be taken to be graded as well.
Minimal decompositions and uniqueness
In this section, all modules will be finitely generated over a Noetherian ring R.A primary decomposition of a submodule M of a module N is called minimal if it has the smallest possible number of primary modules. For minimal decompositions, the primes of the primary modules are uniquely determined: they are the associated primes of N/M. Moreover, the primary submodules associated to the minimal or isolated associated primes are also unique. However the primary submodules associated to the non-minimal associated primes need not be unique.
Example: Let N = R = k for some field k, and let M be the ideal. Then M has two different minimal primary decompositions
M = ∩ = ∩.
The minimal prime is and the embedded prime is.
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 RP 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 Pn of a prime P has a primary decomposition, then its P-primary component is the n-th symbolic power of P.