Section 2.5 of Ideals, Varieties, and Algorithms shows that $G = \{ x + z, y - z \}$ is a Groebner basis for lex order.

  • Dividing $xy$ by $x + z, y - z$ shows that $xy = y(x + z) - z(y - z) - z^2$.
  • Dividing $xy$ by $y - z, x + z$ shows that $xy = x(y - z) + z(x + z) - z^2$.

Thus the remainder is the same, but the “quotients” are different. Therefore, the uniqueness of the remainder is the best one we can hope for from a Groebner basis.