primary decomposition theorem

The primary decomposition theoremPlanetmathPlanetmath for ideals in a given commutative ring (with 1) is a generalizationPlanetmathPlanetmath of the fundamental theorem of arithmeticMathworldPlanetmath. The full statement of the theorem is as follows:

Theorem 1.

Every decomposable ideal in a commutative ring R with 1 has a unique minimal primary decomposition. In other words, if I is an ideal of R with two minimal primary decompositions


then m=n, and after some rearrangement, rad(Ji)=rad(Ki).

The theorem says, that, the number of primary components of a minimal primary decomposition of an ideal, as well as the set of prime radicals associated with the primary components, are unique. This is not to say, however, that the ideal has a unique minimal primary decomposition. For example, let k be a field. Consider the ring k[x,y] of polynomialsMathworldPlanetmathPlanetmath over k in two variables. The ideal (x2,xy) has minimal primary decompositions (x)(x2,y+rx) for every rk.

Remark. To tie the fundamental theorem of arithmetic with this theorem, we observe that every natural numberMathworldPlanetmath n greater than 1 can be uniquely expressed as a productMathworldPlanetmathPlanetmath of prime powers:


where each pi is a prime numberMathworldPlanetmath. This is the same as saying that


as every (pa) is a (p)-primary idealMathworldPlanetmath of for every prime p. The decomposition is minimalPlanetmathPlanetmath, as (p1a1)=(p2a2) iff p1=p2 and a1=a2.


  • 1 D.G. Northcott, Ideal Theory, Cambridge University Press, 1953.
Title primary decomposition theorem
Canonical name PrimaryDecompositionTheorem1
Date of creation 2013-03-22 18:19:56
Last modified on 2013-03-22 18:19:56
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Theorem
Classification msc 13C99