Computing Gröbner basis in $R^n$

Example 11   Let $S$ be the ring of polynomials $Q [x,y]$. Obtain the Gröbner basis of the $S$-submodule of $S^3$ generated by the vectors

$$(x-1,y-1,z-1), (xy-1,yz-2,zx-3).$$

%% gbvec.sm1

[ (x,y,z) ring_of_polynomials [[(x) 1 (y) 1 (z) 1]] weight_vector 0]
define_ring

[ [(x-1).        (y-1).         (z-1).] homogenize
  [(x y - 1).  (y z - 2). (z x - 3).]   homogenize ] /ff set

[ff] groebner {toVectors dehomogenize} map ::