basis

basis (i,M) -- produce a map (of degree i) from a free k-module to M whose image is the degree i part of the module (or ring) M.
basis M -- produce a map from a free k-module to M whose image is the finite dimensional module (or ring) M.

The field k is the coefficient ring of the ring of M. The degree i may be a multi-degree, represented as a list of integers.

     i1 = R = ZZ/101[a..c]
     
     o1 = R
     
     o1 : PolynomialRing
     
     i2 = f = basis(2,R)
     
     o2 = | a2 ab ac b2 bc c2 |
     
                  1        ZZ 6
     o2 : Matrix R  <--- (---)
                          101
A map of R-modules can be obtained by tensoring.
     i3 = f ** R
     
     o3 = | a2 ab ac b2 bc c2 |
     
                  1       6
     o3 : Matrix R  <--- R
     
     i4 = basis(2, ideal(a,b,c)/ideal(a^2,b^2,c^2))
     
     o4 = | b c 0 |
          | 0 0 c |
          | 0 0 0 |
     
     o4 : Matrix
     
     i5 = basis(R/(a^2-a*b, b^2-c^2, b*c))
     
     o5 = | 1 a ab ac ac2 b c c2 |
     
                            R          1        ZZ 8
     o5 : Matrix (--------------------)  <--- (---)
                   2        2    2             101
                  a  - a b,b  - c ,b c
     
     i6 = S = ZZ/101[x,y,z,Degrees=>{{1,3},{1,4},{1,-1}}]
     
     o6 = S
     
     o6 : PolynomialRing
     
     i7 = basis({7,24}, S)
     
     o7 = | x4y3 |
     
                  1        ZZ 1
     o7 : Matrix S  <--- (---)
                          101

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