On the closed subschemes of a scheme

In this article, using the pullback (fiber product) of schemes we define a natural operation, the so-called multiplication, on the set of closed subschemes of a given scheme in an alternative way. It is well known that the pushout of any two closed immersions of schemes with a fixed source exists in the category of schemes. This result allows us to define a second operation, the so-called addition, on the set of closed subschemes of a scheme. Next, it is proved that these structures naturally provide us contravariant functors from the category of schemes to the category of commutative monoids.