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 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 modules over a principal ideal domain">Module (mathematics)">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.
Primary decomposition of an ideal
Let be a Noetherian commutative ring. An ideal of is called primary if it is a proper ideal and for each pair of elements and in such that is in, either or some power of is in ; equivalently, every zero-divisor in the quotient is nilpotent. The radical of a primary ideal is a prime ideal and is said to be -primary for.Let be an ideal in. Then has an irredundant primary decomposition into primary ideals:
Irredundancy means:
- Removing any of the changes the intersection, i.e. for each we have:.
- The prime ideals are all distinct.
- The set is uniquely determined by, and
- If is a minimal element of the above set, then is uniquely determined by ; in fact, is the pre-image of under the localization map.
The elements of are called the prime divisors of or the primes belonging to. In the language of module theory, as discussed below, the set is also the set of associated primes of the -module. Explicitly, that means that there exist elements in such that
By a way of shortcut, some authors call an associated prime of simply an associated prime of .
- The minimal elements of are the same as the minimal prime ideals containing and are called isolated primes.
- The non-minimal elements, on the other hand, are called the embedded primes.
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 embedded prime
For every positive integer, a primary decomposition in of the ideal isThe associated primes are
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-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 to provide just such a complicated output, while, 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.
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 be a ring and a module over it. By definition, an associated prime is a prime ideal which is the annihilator of a nonzero element of ; that is, for some . Equivalently, a prime ideal is an associated prime of if there is an injection of -modules.
A maximal element of the set of annihilators of nonzero elements of can be shown to be a prime ideal and thus, when is a Noetherian ring, there exists an associated prime of if and only if is nonzero.
The set of associated primes of is denoted by or. Directly from the definition,
- If, then.
- For an exact sequence,.
- If is a Noetherian ring, then where refers to support. Also, the set of minimal elements of is the same as the set of minimal elements of.
such that each quotient is isomorphic to for some prime ideals, each of which is necessarily in the support of. Moreover every associated prime of occurs among the set of primes ; i.e.,
In particular, is a finite set when is finitely generated.
Let be a finitely generated module over a Noetherian ring and a submodule of. Given, the set of associated primes of, there exist submodules such that and
A submodule of is called -primary if. A submodule of the -module is -primary as a submodule if and only if it is a -primary ideal; thus, when, the above decomposition is precisely a primary decomposition of an ideal.
Taking, the above decomposition says the set of associated primes of a finitely generated module is the same as when
Properties of associated primes
Let be a Noetherian ring. Then- The set of zero-divisors on is the same as the union of the associated primes of .
- For the same reason, the union of the associated primes of an -module is exactly the set of zero-divisors on, that is, an element such that the endomorphism is not injective.
- Given a subset, an -module, there exists a submodule such that and.
- Let be a multiplicative subset, an -module and the set of all prime ideals of not intersecting. Then is a bijection. Also,.
- Any prime ideal minimal with respect to containing an ideal is in These primes are precisely the isolated primes.
- A module over has finite length if and only if is finitely generated and consists of maximal ideals.
- Let be a ring homomorphism between Noetherian rings and a -module that is flat over. Then, for each -module,
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, an ideal and a minimal prime over, the pre-image of under the localization map is the smallest -primary ideal containing. Thus, in the setting of preceding theorem, the primary ideal corresponding to a minimal prime is also the smallest -primary ideal containing and is called the -primary component of.
For example, if the power of a prime has a primary decomposition, then its -primary component is the -th symbolic power of.