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 Emanuel Lasker (1905) for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether (1921).
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 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[Note 1] was published by Noether's student Grete Hermann (1926).
[1][2] The decomposition does not hold in general for non-commutative Noetherian rings.
is called primary if it is a proper ideal and for each pair of elements
(note this practice will conflict with the usage in the module theory).
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 k. The primary decomposition in
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 for providing such a complicated output, and, nevertheless, being accessible to hand-written computation.
For computing the primary decomposition, we suppose first that 1 is a greatest common divisor of P and Q.
As I 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.
As the greatest common divisor of P and Q is a constant, the resultant D is not zero, and resultant theory implies that I contains all products of D by a monomial in x, y of degree m + n – 1.
The other primary component contains D. One may prove that if P and Q are sufficiently generic (for example if the coefficients of P and Q are distinct indeterminates), then there is only another primary component, which is a prime ideal, and is generated by P, Q and D. In algebraic geometry, an affine algebraic set V(I) is defined as the set of the common zeros of an ideal I of a polynomial ring
An irredundant primary decomposition of I defines a decomposition of V(I) into a union of algebraic sets V(Qi), which are irreducible, as not being the union of two smaller algebraic sets.
and Lasker–Noether theorem shows that V(I) has a unique irredundant decomposition into irreducible algebraic varieties where the union is restricted to minimal associated primes.
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.
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, there exists an associated prime of M if and only if M is nonzero.
be a finitely generated module over a Noetherian ring R and N a submodule of M. Given
, the above decomposition says the set of associated primes of a finitely generated module M is the same as
Then The next theorem gives necessary and sufficient conditions for a ring to have primary decompositions for its ideals.
[18] 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. 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.