Category of Rings - As A Concrete Category

As A Concrete Category

The category Ring is a concrete category meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions preserving this structure. There is a natural forgetful functor

U : Ring → Set

for the category of rings to the category of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has a left adjoint

F : Set → Ring

which assigns to each set X the free ring generated by X.

One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of monoids). Specifically, there are faithful functors

A : Ring → Ab

M : Ring → Mon

which "forget" multiplication and addition, respectively. Both of these functors have left adjoints. The left adjoint of A is the functor which assigns to every abelian group X (thought of as a Z-module) the tensor ring T(X). The left adjoint of M is the functor which assigns to every monoid X the integral monoid ring Z.