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# tensor(RingMap,Matrix) -- tensor product via a ring map

- Function: tensor
- Usage:
    `f ** M`
    `tensor(f, M)`
- Inputs:
  - f, a ring map, from $R \to S$
  - M, a matrix or a module, over the source ring $R$ of f
- Outputs:
  - a matrix or a module, the same type as M, but over the target ring $S$ of f

## Description

    i1 : R = QQ[a..d]

    o1 = R

    o1 : PolynomialRing

    i2 : S = QQ[s,t]

    o2 = S

    o2 : PolynomialRing

    i3 : F = map(S,R,{s^4,s^3*t,s*t^3,t^4})

                      4   3      3   4
    o3 = map (S, R, {s , s t, s*t , t })

    o3 : RingMap S <-- R

    i4 : m = matrix{{a,b,c,d}}

    o4 = | a b c d |

                 1      4
    o4 : Matrix R  <-- R

    i5 : F ** m

    o5 = | s4 s3t st3 t4 |

                 1      4
    o5 : Matrix S  <-- S

    i6 : F ** image m

    o6 = cokernel {1} | -s3t 0    -st3 0   0   -t4 |
                  {1} | s4   -st3 0    0   -t4 0   |
                  {1} | 0    s3t  s4   -t4 0   0   |
                  {1} | 0    0    0    st3 s3t s4  |

                                4
    o6 : S-module, quotient of S

## See also

- RingMap Module -- apply a ring map
- RingMap Module -- apply a ring map

## Ways to use this method:

- tensor(RingMap,Matrix) -- tensor product via a ring map
- `tensor(RingMap,Module)`

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/functions/tensor-doc.m2:200:0.