The category Ring is a concrete category 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 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 which assigns to each set X the free ring generated by X.
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 forgetful functors to Ab and Mon also create and preserve limits.
A necessary condition for there to be morphisms from R to S is that the characteristic of S divide that of R. Note that even though some of the hom-sets are empty, the category Ring is still connected since it has an initial object.
This category is one of the central objects of study in the subject of commutative algebra.
Any ring can be made commutative by taking the quotient by the ideal generated by all elements of the form (xy − yx).
The free commutative ring on a set of generators E is the polynomial ring Z[E] whose variables are taken from E. This gives a left adjoint functor to the forgetful functor from CRing to Set.
The equivalence is given by the contravariant functor Spec which sends a commutative ring to its spectrum, an affine scheme.
The connected components of Field are the full subcategories of characteristic p, where p = 0 or is a prime number.
This functor has a left adjoint which sends each ring A to the tensor product R⊗ZA, thought of as an R-algebra by setting r·(s⊗a) = rs⊗a.
For distinction we call such algebraic structures rngs and their morphisms rng homomorphisms.
The inclusion functor Ring → Rng respects limits but not colimits.