- Category of rings
In

mathematics , the**category of rings**, denoted by**Ring**, is the category whose objects are rings (with identity) and whosemorphism s arering homomorphism s (preserving the identity). Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.**As a concrete category**The category

**Ring**is aconcrete category meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions preserving this structure. There is a naturalforgetful functor :"U" :**Ring**→**Set**for the category of rings to thecategory of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has aleft adjoint :"F" :**Set**→**Ring**which assigns to each set "X" thefree ring generated by "X".One can also view the category of rings as a concrete category over

**Ab**(thecategory of abelian groups ) or over**Mon**(thecategory of monoids ). Specifically, there arefaithful functor s:"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 everyabelian group "X" (thought of as a**Z**-module) thetensor ring "T"("X"). The left adjoint of "M" is the functor which assigns to everymonoid "X" the integralmonoid ring **Z**["M"] .**Properties****Limits and colimits**The category

**Ring**is both complete and cocomplete, meaning that all smalllimits and colimits exist in**Ring**. Like many other algebraic categories, the forgetful functor "U" :**Ring**→**Set**creates (and preserves) limits andfiltered colimit s, but does not preserve eithercoproduct s orcoequalizer s. The forgetful functors to**Ab**and**Mon**also create and preserve limits.Examples of limits and colimits in

**Ring**include:*The ring of

integer s**Z**forms aninitial object in**Ring**.

*Anytrivial ring (i.e. a ring with a single element 0 = 1) forms aterminal object .

*The product in**Ring**is given by thedirect product of rings . This is just thecartesian product of the underlying sets with addition and multiplication defined component-wise.

*Thecoproduct of a family of rings exists and is given by a construction analogous to thefree product of groups. It's quite possible for the coproduct of nontrivial rings to be trivial. In particular, this happens whenever the factors haverelatively prime characteristic (since the characteristic of the coproduct of ("R"_{"i"})_{"i"∈"I"}must divide the characteristics of each of the rings "R"_{"i"}).

*Theequalizer in**Ring**is just the set-theoretic equalizer (the equalizer of two ring homomorphisms is always asubring ).

*Thecoequalizer 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" thekernel pair of "f" (this is just the pullback of "f" with itself) is acongruence relation on "R". The ideal determined by this congruence relation is precisely the (ring-theoretic) kernel of "f". Note thatcategory-theoretic kernel s do not make sense in**Ring**since there are nozero morphism s (see below).

*The ring of "p"-adic integers is theinverse limit in**Ring**of a sequence of rings of integers mod "p"^{"n"}**Morphisms**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 thetrivial ring 0 to any nontrivial 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:*

Isomorphism s in**Ring**are thebijective ring homomorphisms.

*Monomorphism s in**Ring**are theinjective homomorphisms. Not every monomorphism is regular however.

*Every surjective homomorphism is anepimorphism in**Ring**, but the converse is not true. The inclusion**Z**→**Q**is a nonsurjective epimorphism. The natural ring homomorphism from any commutative ring "R" to any one of its localizations is an epimorphism which is not necessarily surjective.

*The surjective homomorphisms can be characterized as the regular orextremal epimorpism s in**Ring**(these two classes coinciding).

*Bimorphism s 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 s in**Ring**are the trivial rings (i.e. the terminal objects).

*Lackingzero morphism s, the category of rings cannot be apreadditive category . (However, every ring—considered as a small category with a single object— is a preadditive category).

*The category of rings is asymmetric monoidal category with thetensor product of rings ⊗_{Z}as the monoidal product and the ring of integers**Z**as the unit object. It follows from theEckmann–Hilton theorem , that a monoid in**Ring**is just acommutative ring . The action of a monoid (= commutative ring) "R" on a object (= ring) "A" of**Ring**is just a "R"-algebra.**ubcategories**The category of rings has a number of important

subcategories . These include thefull subcategories ofcommutative rings ,integral domain s,principal ideal domain s, and fields.**Category of commutative rings**The

**category of commutative rings**, denoted**CRing**, is the full subcategory of**Ring**whose objects are allcommutative rings . This category is one of the central objects of study in the subject ofcommutative algebra .Any ring can be made commutative by taking the quotient by the ideal generated by all elements of the form ("xy" − "yx"). This defines a functor

**Ring**→**CRing**which is left adjoint to the inclusion functor, so that**CRing**is areflective subcategory of**Ring**. Thefree commutative ring on a set of generators "E" is thepolynomial ring **Z**["E"] whose variables are taken from "E". This gives a left adjoint functor to the the forgetful functor from**CRing**to**Set**.**CRing**is limit-closed in**Ring**, which means that limits in**CRing**are the same as they are in**Ring**. Colimits, however, are generally different. They can be formed by taking the commutative quotient of colimits in**Ring**. The coproduct of two commutative rings is given by thetensor product of rings . Again, its quite possible for the coproduct of two nontrivial commutative rings to be trivial.The

opposite category of**CRing**is equivalent to thecategory of affine schemes . The equivalence is given by thecontravariant functor Spec which sends a commutative ring to its spectrum, an affine scheme.**Category of fields**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 finitely complete nor finitely cocomplete. In particular,

**Field**has neither products nor coproducts.Another curious aspect of the category of fields is that every morphism is a

monomorphism . This follows from the fact that the only ideals in a field "F" are thezero ideal and "F" itself. One can then view morphisms in**Field**asfield extension s.The category of fields is not connected. There are no morphisms between fields of different characteristic. The connected components of

**Field**are the full subcategories of characteristic "p", where "p" = 0 or is aprime number . Each such subcategory has aninitial object : theprime field of characteristic "p" (which is**Q**if "p" = 0, otherwise thefinite field **F**_{"p"}).**Related categories and functors****Category of groups**There is a natural functor from

**Ring**to thecategory of groups ,**Grp**, which sends each ring "R" to itsgroup of units "U"("R") and each ring homomorphism to the restriction to "U"("R"). This functor has aleft adjoint which sends each group "G" to theintegral group ring **Z**["G"] .Another functor between these categories is provided by the group "G(R)" of projectivities generated by an associative ring through

inversive ring geometry .**"R"-algebras**Given a commutative ring "R" one can define the category

**"R"-Alg**whose objects are all "R"-algebras and whose morphisms are "R"-algebra homomorphisms.The category of rings can be considered a special case. Every ring can be considered a

**Z**-algebra is a unique way. Ring homomorphisms are precisely the**Z**-algebra homomorphisms. The category of rings is, therefore, isomorphic to the category**Z-Alg**. 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 tensor product "R"⊗_{Z}"A", thought of as an "R"-algebra by setting "r"·("s"⊗"a") = "rs"⊗"a".**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**. 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 a identity to any rng. This makes**Ring**into a (nonfull)reflective subcategory of**Rng**.The

trivial ring serves as both a initial and terminal object in**Rng**(that is, it is azero object ). It follows that**Rng**, like**Grp**but unlike**Ring**, haszero morphism s. These are just the rng homomorphisms that map everything to 0. Despite the existence of zero morphisms,**Rng**is still not apreadditive category . The addition of two rng homomorphism (computed pointwise) is generally not a rng homomorphism.Limits in

**Rng**are generally the same as in**Ring**, but colimits can take a different form. In particular, thecoproduct of two rngs is given by adirect sum construction analogous to that of abelian groups.Free constructions are less natural in

**Rng**then they are in**Ring**. For example, the free rng generated by a set {"x"} is the rng of all integral polynomials over "x" with no constant term, while the free ring generated by {"x"} is just thepolynomial ring **Z**["x"] .**References***cite book | last = Adámek | first = Jiří | coauthors = Horst Herrlich, and George E. Strecker | year = 1990 | url = http://katmat.math.uni-bremen.de/acc/acc.pdf | title = Abstract and Concrete Categories | publisher = John Wiley & Sons | id = ISBN 0-471-60922-6

*cite book | first = Saunders | last = Mac Lane | authorlink = Saunders Mac Lane | coauthors =Garrett Birkhoff | title = Algebra | edition = (3rd ed.) | publisher = American Mathematical Society | location = Providence, Rhode Island | year = 1999 | isbn = 0-8218-1646-2

*cite book | first = Saunders | last = Mac Lane | year = 1998 | title =Categories for the Working Mathematician | series = Graduate Texts in Mathematics**5**| edition = (2nd ed.) | publisher = Springer | id = ISBN 0-387-98403-8

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