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# primaryComponent -- find a primary component corresponding to an associated prime

- Usage:
    `Q = primaryComponent(I, P)`
- Inputs:
  - I, an ideal, an ideal in a (quotient of a) polynomial ring R
  - P, an ideal, an associated prime of I
- Optional inputs:
  - Increment => ..., default value 1
  - Strategy => ..., default value 2
- Outputs:
  - Q, an ideal, a P-primary ideal of I

## Description

The output Q is topComponents(I + P^m) for sufficiently large m. The criterion that Q is primary is given in Eisenbud-Huneke-Vasconcelos, Invent. Math. 110 (1992) 207-235. However, we use localize(Ideal,Ideal).

The Strategy option value sets the strategy option for localize, and should be one of the following:

- Strategy => 0 -- Uses localize Strategy 0
- Strategy => 1 -- Uses localize Strategy 1
- Strategy => 2 -- Uses localize Strategy 2

The Increment option value should be an integer. The algorithm given in Eisenbud-Huneke-Vasconcelos, Invent. Math. 110 (1992) 207-235, relies on topComponents(I + P^m) for $m$ sufficiently large. The algorithm begins with $m = 1$, and increases $m$ by the value of the Increment option until m is sufficiently large. The default value is 1.

Authored by C. Yackel. Last modified June, 2000.

## See also

- associatedPrimes(Ideal) -- find associated primes
- primaryDecomposition(Ideal) -- irredundant primary decomposition of an ideal
- radical -- the radical of an ideal
- minimalPrimes -- minimal primes of an ideal
- topComponents -- compute top dimensional component of an ideal or module
- removeLowestDimension -- remove components of lowest dimension

## Ways to use primaryComponent:

- `primaryComponent(Ideal,Ideal)`

## For the programmer

The object primaryComponent is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PrimaryDecomposition/doc.m2:750:0.