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Concept
Category of rings
Summary
In mathematics, the category of rings, denoted by Ring, is the whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is , meaning that the class of all rings is proper.
The category Ring is a meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure. There is a natural forgetful functor
U : Ring → Set
for the category of rings to the 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 ) or over Mon (the ). Specifically, there are forgetful 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[X].
The category Ring is both , meaning that all small limits and colimits exist in Ring. Like many other algebraic categories, the forgetful functor U : Ring → Set creates (and preserves) limits and filtered colimits, but does not preserve either coproducts or coequalizers. The forgetful functors to Ab and Mon also create and preserve limits.
Examples of limits and colimits in Ring include:
The ring of integers Z is an initial object in Ring.
The zero ring is a terminal object in Ring.
The in Ring is given by the direct product of rings. This is just the cartesian product of the underlying sets with addition and multiplication defined component-wise.
The coproduct of a family of rings exists and is given by a construction analogous to the free product of groups.
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