In mathematics, the LaskerâÂÂNoether 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 LaskerâÂÂNoether 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.
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:
Moreover, this decomposition is unique in the two ways:
Primary ideals which correspond to non-minimal prime ideals over 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 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 (note this practice will conflict with the usage in the module theory).
In the case of the ring of integers , the LaskerâÂÂNoether theorem is equivalent to the fundamental theorem of arithmetic. If an integer 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
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 .
The primary decomposition in of the ideal 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
In , the ideal is a primary ideal that has as associated prime. It is not a power of its associated prime.
For every positive integer , a primary decomposition in of the ideal is
The associated primes are
Example: Let N = R = k[x, y] for some field k, and let M be the ideal (xy, y<sup>2</sup>). Then M has two different minimal primary decompositions M = (y) ∩ (x, y<sup>2</sup>) = (y) ∩ (x + y, y<sup>2</sup>). The minimal prime is (y) and the embedded prime is (x, y).
In the ideal has the (non-unique) primary decomposition
The associated prime ideals are and is a non associated prime ideal such that
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 (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 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 (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 .
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.
If is the associated prime of , then and LaskerâÂÂNoether 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.
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 (this implies ). 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 is a finitely generated module over , then there is a finite ascending sequence of submodules
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 general, these inclusions are not the equalities.) 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 (without finite generation, there can be infinitely many associated primes.)
Let be a Noetherian ring. Then
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 AtiyahâÂÂMacdonald 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 .
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.