primary decomposition

Let R be a commutative ring and A be an ideal in R. A decomposition of A is a way of writing A as a finite intersectionMathworldPlanetmath of primary idealsMathworldPlanetmath:


where the Qi are primary in R.

Not every ideal admits a primary decomposition, so we define a decomposable ideal to be one that does.

Example. Let R= and take A=(180). Then A is decomposable, and a primary decomposition of A is given by


since (4), (9), and (5) are all primary ideals in .

Given a primary decomposition A=Qi, we say that the decomposition is a minimal primary decomposition if for all i, the prime idealsMathworldPlanetmathPlanetmathPlanetmath Pi=rad(Qi) (where rad denotes the radicalPlanetmathPlanetmathPlanetmath of an ideal) are distinct, and for all 1in, we have


In the example above, the decomposition (4)(9)(5) of A is minimalPlanetmathPlanetmath, where as A=(2)(4)(3)(9)(5) is not.

Every primary decomposition can be refined to admit a minimal primary decomposition.

Title primary decomposition
Canonical name PrimaryDecomposition
Date of creation 2013-03-22 14:15:05
Last modified on 2013-03-22 14:15:05
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Definition
Classification msc 13C99
Synonym shortest primary decomposition
Defines decomposable ideal
Defines minimal primary decomposition