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# modulo -- find the pre-image (pullback) of image of a map (low level version)

- Usage:
    `modulo(f,g)`
- Inputs:
  - f
  - g
- Outputs:
  - a matrix, whose image is the pre-image (pullback) of the image of g under f

## Description

The maps f and g must have the same target, and their sources and targets must be free. If f is null, then it is taken to be the identity. If g is null, it is taken to be zero.

This function is mainly for internal use.

    i1 : R = QQ[x,y,z]

    o1 = R

    o1 : PolynomialRing

    i2 : f = matrix {{x,y}}

    o2 = | x y |

                 1      2
    o2 : Matrix R  <-- R

    i3 : g = matrix {{y,z}}

    o3 = | y z |

                 1      2
    o3 : Matrix R  <-- R

    i4 : modulo(f,g)

    o4 = {1} | 0 z y |
         {1} | 1 0 0 |

                 2      3
    o4 : Matrix R  <-- R

    i5 : kernel( inducedMap(coker g, target g) * f )

    o5 = image {1} | 0 z y |
               {1} | 1 0 0 |

                                 2
    o5 : R-module, submodule of R

## Ways to use modulo:

- `modulo(Matrix,Matrix)`
- `modulo(Matrix,Nothing)`
- `modulo(Nothing,Matrix)`

## For the programmer

The object modulo is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/functions/modulo-doc.m2:27:0.