Category of rings

In

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 that preserve this structure. There is a natural forgetful functor

U : Ring → Set

for the category of rings to the

F : Set → Ring

which assigns to each set X the

One can also view the category of rings as a concrete category over Ab (the

• The ring of
  initial object
  in Ring.
• The
  terminal object
  in Ring.
• The
  direct product of rings. This is just the cartesian product
  of the underlying sets with addition and multiplication defined component-wise.
• The
  relatively prime characteristic
  (since the characteristic of the coproduct of (R_i)_i∈I must divide the characteristics of each of the rings R_i).
• The
  equalizer in Ring is just the set-theoretic equalizer (the equalizer of two ring homomorphisms is always a subring
  ).
• The coequalizer of two ring homomorphisms f and g from R to S is the quotient of S by the ideal generated by all elements of the form f(r) − g(r) for r ∈ R.
• Given a ring homomorphism f : R → S the
  category-theoretic kernels do not make sense in Ring since there are no zero morphisms
  (see below).

Unlike many categories studied in mathematics, there do not always exist morphisms between pairs of objects in Ring. This is a consequence of the fact that ring homomorphisms must preserve the identity. For example, there are no morphisms from the zero ring 0 to any nonzero ring. 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.

Some special classes of morphisms in Ring include:

• bijective
  ring homomorphisms.
• regular
  however.
• Every surjective homomorphism is an
  localizations
  is an epimorphism which is not necessarily surjective.
• The surjective homomorphisms can be characterized as the
  extremal epimorphisms
  in Ring (these two classes coinciding).
• Bimorphisms
  in Ring are the injective epimorphisms. The inclusion Z → Q is an example of a bimorphism which is not an isomorphism.

Other properties

• The only injective object in Ring up to isomorphism is the zero ring (i.e. the terminal object).
• Lacking zero morphisms, the category of rings cannot be a preadditive category. (However, every ring—considered as a category with a single object—is a preadditive category).
• The category of rings is a
  R-algebra
  .

Subcategories

The category of rings has a number of important

The category of fields, denoted Field, is the full subcategory of CRing whose objects are fields. The category of fields is not nearly as well-behaved as other algebraic categories. In particular, free fields do not exist (i.e. there is no left adjoint to the forgetful functor Field → Set). It follows that Field is not a reflective subcategory of CRing.

The category of fields is neither

Another curious aspect of the category of fields is that every morphism is a

Another functor between these categories sends each ring R to the group of units of the matrix ring M₂(R) which acts on the projective line over a ring P(R).

R-algebras

Given a commutative ring R one can define the category R-Alg whose objects are all

The category of rings can be considered a special case. Every ring can be considered a Z-algebra in a unique way. Ring homomorphisms are precisely the Z-algebra homomorphisms. The category of rings is, therefore, isomorphic to the category Z-Alg.^[1] Many statements about the category of rings can be generalized to statements about the category of R-algebras.

For each commutative ring R there is a functor R-Alg → Ring which forgets the R-module structure. This functor has a left adjoint which sends each ring A to the

Rings without identity

Many authors do not require rings to have a multiplicative identity element and, accordingly, do not require ring homomorphism to preserve the identity (should it exist). This leads to a rather different category. For distinction we call such algebraic structures rngs and their morphisms rng homomorphisms. The category of all rngs will be denoted by Rng.

The category of rings, Ring, is a nonfull subcategory of Rng. It is nonfull because there are rng homomorphisms between rings which do not preserve the identity, and are therefore not morphisms in Ring. The inclusion functor Ring → Rng has a left adjoint which formally adjoins an identity to any rng. The inclusion functor Ring → Rng respects limits but not colimits.

The

There is a fully faithful functor from the category of abelian groups to Rng sending an abelian group to the associated rng of square zero.

Free constructions are less natural in Rng than they are in Ring. For example, the free rng generated by a set {x} is the ring of all integral polynomials over x with no constant term, while the free ring generated by {x} is just the polynomial ring Z[x].

References