Category of rings

# Category of rings

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the category of rings, denoted by Ring, is the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

whose objects are rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

(with identity) and whose morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

s are ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....

s (preserving the identity). Like many categories in mathematics, the category of rings is large, meaning that the class
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...

of all rings is proper.

## As a concrete category

The category Ring is a concrete category
Concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...

meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s preserving this structure. There is a natural forgetful functor
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...

U : RingSet

for the category of rings to the category of sets
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...

which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has a left adjoint
F : SetRing

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
Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....

) or over Mon (the category of monoids). Specifically, there are faithful functors
A : RingAb
M : RingMon

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
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

X (thought of as a Z-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

) the tensor ring T(X). The left adjoint of M is the functor which assigns to every monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

X the integral monoid ring Z[M].

### Limits and colimits

The category Ring is both complete and cocomplete
Complete category
In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C where J is small has a limit in C. Dually, a cocomplete category is one in which all small colimits exist...

, meaning that all small limits and colimits exist in Ring. Like many other algebraic categories, the forgetful functor U : RingSet creates (and preserves) limits and filtered colimits, but does not preserve either coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...

s or coequalizer
Coequalizer
In category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category...

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
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s Z forms an initial object
Initial object
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...

in Ring.
• Any trivial ring (i.e. a ring with a single element 0 = 1) forms a terminal object.
• The product
Product (category theory)
In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...

in Ring is given by the direct product of rings. This is just the cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

of the underlying sets with addition and multiplication defined component-wise.
• The coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...

of a family of rings exists and is given by a construction analogous to the free product
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...

of groups. It's quite possible for the coproduct of nontrivial rings to be trivial. In particular, this happens whenever the factors have relatively prime characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...

(since the characteristic of the coproduct of (Ri)iI must divide the characteristics of each of the rings Ri).
• The equalizer
Equalizer
Equalizer or equaliser may refer to:*Equalization, the process of adjusting the strength of certain frequencies within a signal*An equalization filter for used audio and similar signals...

in Ring is just the set-theoretic equalizer (the equalizer of two ring homomorphisms is always a subring
Subring
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...

).
• The coequalizer
Coequalizer
In category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category...

of two ring homomorphisms f and g from R to S is the quotient
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...

of S by the ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

generated by all elements of the form f(r) − g(r) for rR.
• Given a ring homomorphism f : RS the kernel pair of f (this is just the pullback
Pullback (category theory)
In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan X \rightarrow Z \leftarrow Y...

of f with itself) is a congruence relation
Congruence relation
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...

on R. The ideal determined by this congruence relation is precisely the (ring-theoretic) kernel of f. Note that category-theoretic kernels do not make sense in Ring since there are no zero morphism
Zero morphism
In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...

s (see below).
• The ring of p-adic integers is the inverse limit
Inverse limit
In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...

in Ring of a sequence of rings of integers mod pn
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

### 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 the trivial ring 0 to any nontrivial ring. A necessary condition for there to be morphisms from R to S is that the characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...

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
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

s in Ring are the bijective ring homomorphisms.
• Monomorphism
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....

s in Ring are the injective homomorphisms. Not every monomorphism is regular however.
• Every surjective homomorphism is an epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...

in Ring, but the converse is not true. The inclusion ZQ is a nonsurjective epimorphism. The natural ring homomorphism from any commutative ring R to any one of its localizations
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...

is an epimorphism which is not necessarily surjective.
• The surjective homomorphisms can be characterized as the regular or extremal epimorphisms in Ring (these two classes coinciding).
• Bimorphisms in Ring are the injective epimorphisms. The inclusion ZQ is an example of a bimorphism which is not an isomorphism.

### Other properties

• The only injective object
Injective object
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories...

s in Ring are the trivial rings (i.e. the terminal objects).
• Lacking zero morphism
Zero morphism
In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...

s, the category of rings cannot be a preadditive category
In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...

. (However, every ring—considered as a small category with a single object— is a preadditive category).
• The category of rings is a symmetric monoidal category with the tensor product of rings ⊗Z as the monoidal product and the ring of integers Z as the unit object. It follows from the Eckmann–Hilton theorem, that a monoid
Monoid (category theory)
In category theory, a monoid in a monoidal category is an object M together with two morphisms* \mu : M\otimes M\to M called multiplication,* and \eta : I\to M called unit,...

in Ring is just a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

. The action of a monoid (= commutative ring) R on an object (= ring) A of Ring is just a R-algebra.

## Subcategories

The category of rings has a number of important subcategories. These include the full subcategories of commutative rings, integral domains, principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...

s, and field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

s.

### Category of commutative rings

The category of commutative rings, denoted CRing, is the full subcategory of Ring whose objects are all commutative rings. This category is one of the central objects of study in the subject of commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

.

Any ring can be made commutative by taking the quotient
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...

by the ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

generated by all elements of the form (xyyx). This defines a functor RingCRing which is left adjoint to the inclusion functor, so that CRing is a reflective subcategory
Reflective subcategory
In mathematics, a subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector...

of Ring. The free commutative ring on a set of generators E is the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

Z[E] whose variables are taken from E. This gives a left adjoint functor to 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 the tensor product of rings. Again, it's quite possible for the coproduct of two nontrivial commutative rings to be trivial.

The opposite category
Opposite category
In category theory, a branch of mathematics, the opposite category or dual category Cop of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite...

of CRing is equivalent
Equivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics...

to the category of affine schemes. The equivalence is given by the contravariant functor Spec which sends a commutative ring to its spectrum
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...

, an affine scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

.

### Category of fields

The category of fields, denoted Field, is the full subcategory of CRing whose objects are field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

s. 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 FieldSet). 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
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....

. This follows from the fact that the only ideals in a field F are the zero ideal and F itself. One can then view morphisms in Field as field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...

s.

The category of fields is not connected. There are no morphisms between fields of different characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...

. The connected components of Field are the full subcategories of characteristic p, where p = 0 or is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

. Each such subcategory has an initial object
Initial object
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...

: the prime field of characteristic p (which is Q if p = 0, otherwise the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

Fp).

### Category of groups

There is a natural functor from Ring to the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...

, Grp, which sends each ring R to its group of units U(R) and each ring homomorphism to the restriction to U(R). This functor has a left adjoint which sends each group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

G to the integral 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
Inversive ring geometry
In mathematics, inversive ring geometry is the extension of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio to the context of associative rings, concepts usually built upon rings that happen to be fields....

.

A functor from the category of abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s is the one that sends each abelian group in the corresponding zero ring
Zero ring
In ring theory, a branch of mathematics, a zero ring is a ring in which the product of any two elements is 0 . In ring theory, a branch of mathematics, a zero ring is a ring (without unity) in which the product of any two elements is 0 (the additive identity element). In ring theory, a branch of...

.

### R-algebras

Given a commutative ring R one can define the category R-Alg whose objects are all R-algebras
Algebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....

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
Isomorphism of categories
In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D and GF = 1C. This means that both the objects and the morphisms of C and D stand in a one to one correspondence to each other...

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 RZA, thought of as an R-algebra by setting r·(sa) = rsa.

### 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
Rng (algebra)
In abstract algebra, a rng is an algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element...

and their morphisms rng homomorphisms. The category of all rngs will be denoted by Rng.

The category of rings, Ring, is a nonfull subcategory
Subcategory
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and...

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 RingRng has a left adjoint which formally adjoins an identity to any rng. This makes Ring into a (nonfull) reflective subcategory
Reflective subcategory
In mathematics, a subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector...

of Rng.

The trivial ring serves as both an initial and terminal object in Rng (that is, it is a zero object). It follows that Rng, like Grp but unlike Ring, has zero morphism
Zero morphism
In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...

s. These are just the rng homomorphisms that map everything to 0. Despite the existence of zero morphisms, Rng is still not a preadditive category
In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...

. 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, the coproduct
Coproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...

of two rngs is given by a direct sum
Direct sum
In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...

construction analogous to that of abelian groups.

Free construction
Free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms....

s 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 the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

Z[x].