How to create homomorphism from the free module ZZ^4 to ZZ[i] , say f(x,y,z,w)= x-iy iz-w and compute its kernel

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Like this:

sage: A = ZZ^4; B = ZZ[I]
sage: H = Hom(A,B)
sage: f = H( [B.one(), -I, I, B.one()] ); f
Free module morphism defined by the matrix
[ 1  0]
[ 0 -1]
[ 0  1]
[ 1  0]
Domain: Ambient free module of rank 4 over the principal ideal domain Integer Ring
Codomain: Gaussian Integers generated by I0 in Number Field in I0 with defining polynomial x^2 + 1 with I0 = 1*I
sage: f.kernel()
Free module of degree 4 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1  0  0 -1]
[ 0  1  1  0]

About that morphism $$F(z,w,s,t)=2iz-iw+3\sqrt3s-\sqrt3t\ ,$$ seen as a morphism of abelian groups, it can be defined as follows as a morphism from $\Bbb Z^4$ to the (abelian group obained by applying the forgetful functor from the) field $\Bbb Q(\sqrt{-1},\sqrt 3)$ as follows, using the short cuts $j=\sqrt{-1}$ and $b=\sqrt 3$:

A = ZZ^4
F.<j,b> = NumberField([sqrt(-1).minpoly(), sqrt(3).minpoly()])

f = A.Hom(F^1)((2*j, -j, 3*b, -b))

Then we have for instance:

sage: f((1,1,1,1))
(j + 2*b)

The "trick" is best explained by examining the following objects:

sage: F
Number Field in j with defining polynomial x^2 + 1 over its base field
sage: F^1
Vector space of dimension 1 over Number Field in j with defining polynomial x^2 + 1 over its base field