z-plane in . This shows how the ideal quotient can be used to "delete" irreducible subschemes. A useful scheme theoretic example is taking the ideal quotient of a reducible ideal. For example, the ideal quotient , showing that the ideal quotient of a subscheme of some non-reduced scheme, where both have the same reduced subscheme, kills off some of the non-reduced structure. We can use the previous example to find the saturation of an ideal corresponding to a projective scheme. Given a homogeneous ideal the saturation of is defined as the ideal quotient where . It is a theorem that the set of saturated ideals of contained in is in bijection with the set of projective subschemes in . This shows us that defines the same projective curve as in . References Viviana Ene, Jürgen Herzog: 'Gröbner Bases in Commutative Algebra', AMS Graduate Studies in Mathematics, Vol 130 (AMS 2012) M.F.Atiyah, I.G.MacDonald: 'Introduction to Commutative Algebra', Addison-Wesley 1969. Ideals (ring theory)