Lasker–Noether theorem

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In mathematics, the Lasker–Noether theorem and states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection 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 the world chess champion Emanuel Lasker (1905) for the special case of polynomial rings, and was proven in its full generality by Emmy Noether (1921). 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.

The Lasker-Noether theorem is an extension of the fundamental theorem of arithmetic to all Noetherian rings. It is also an extension of the structure theorem for finitely generated abelian groups to finitely generated modules over a Noetherian ring. 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 was published by Noether's student Grete Hermann (1926).

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[edit] 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 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 element of M.

[edit] Statement

The Lasker-Nother theorem for modules states every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules. For the special case of ideals it states that every ideal of a Noetherian ring is a finite intersection of primary ideals.

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.

[edit] 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 associated primes (those not containing any other associated primes) are also unique. Hoever the primary submodules associated to the non-miminal associated primes (called embedded primes for geometric reasons) need not be unique.

Example: Let N = R = k[x, y] for some field k, and let M be the ideal (xy, y2). Then M has two different minimal primary decompositions M = (y) ∩ (x, y2) = (y) ∩ (x + y, y2). The minimal prime is (y) and the embedded prime is (x, y).

[edit] Primary ideals

If Q is a primary ideal, then the associated prime ideal P is the radical of Q.

If P is a maximal prime ideal, then any ideal containing a power of P is P-primary. Not all P-primary ideals need be powers of P; for example the ideal (x,y2) is P-primary for the ideal P=(x,y) in the ring k[x,y], but is not a power of P.

In general powers of a prime ideal P need not be P-primary. (An example is given by taking R to be the ring k[x,y,z]/(xy−z2), with P the prime ideal (x,z). If Q=P2, then xy∈Q, but x is not in Q and y is not in the radical P of Q, so Q is not P-primary.) However every ideal Q with radical P is contained in a smallest P-primary ideal, consisting of all elements a such that ax is in Q for some x not in P. In particular there is a smallest P-primary ideal containing Pn, called the nth symbolic power of P.

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