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# isPrimary -- determine whether a submodule is primary

- Usage:
    `isPrimary Q`
    `isPrimary(Q, P)`
    `isPrimary(M, Q)`
- Inputs:
  - Q, an ideal or a module, the submodule or ideal to be checked for being primary
  - P, an ideal, the radical of Q
  - M, a module, the ambient module
- Optional inputs:
  - Strategy => a thing, default value null, See associatedPrimes(...,Strategy=>...) and isPrime(Ideal, Strategy => ...) (missing documentation)
- Outputs:
  - a Boolean value, true if Q is primary, false otherwise

## Description

Checks to see if a given submodule Q of a module M is primary, i.e. whether or not M/Q has exactly one associated prime (which is equivalent for finitely generated modules over Noetherian rings). If the input is a single ideal, then the ambient module is taken to be the ring (i.e. the free module of rank 1), and does not need to be specified.

    i1 : Q = ZZ/101[x,y,z]

    o1 = Q

    o1 : PolynomialRing

    i2 : isPrimary ideal(y^6)

    o2 = true

    i3 : isPrimary(ideal(y^6), ideal(y))

    o3 = true

    i4 : isPrimary ideal(x^4, y^7)

    o4 = true

    i5 : isPrimary ideal(x*y, y^2)

    o5 = false

## See also

- primaryDecomposition(Ideal) -- irredundant primary decomposition of an ideal
- primaryDecomposition(Module) -- irredundant primary decomposition of a module
- associatedPrimes -- find associated primes

## Ways to use isPrimary:

- `isPrimary(Ideal)`
- `isPrimary(Ideal,Ideal)`
- `isPrimary(Module,Module)`

## For the programmer

The object isPrimary 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.