  In this paper, we obtain some new results on closed subschemes. Specially, we
define natural addition and multiplication on the closed subschemes of a
scheme. It is shown that "the multiplication" precisely coincides with the well
known notion of "the scheme-theoretic intersection". Dually, "the addition"
coincides with "the scheme-theoretic union". It is also proved that these
structures naturally provide contravariant functors from the category of
schemes to the category of commutative monoids.
