mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the category of rings, denoted by Ring, is the

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

whose objects are rings (with identity) and whose

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

s are

ring homomorphism In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...

s (that preserve the identity). Like many categories in mathematics, the category of rings is

large Large means of great size. Large may also refer to: Mathematics * Arbitrarily large, a phrase in mathematics * Large cardinal, a property of certain transfinite numbers * Large category, a category with a proper class of objects and morphisms (o ...

, meaning that the

class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...

of all rings is proper.

As a concrete category

The category Ring is a

concrete category In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of the objects of the category as sets with additional ...

meaning that the objects are

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

s with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure. There is a natural

forgetful functor In mathematics, more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's structure or properties mapping to the output. For an algebraic structure of ...

:''U'' : Ring → Set for the category of rings to the

category of sets In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...

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

left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

:''F'' : Set → Ring which assigns to each set ''X'' the

free ring In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the p ...

generated by ''X''. One can also view the category of rings as a concrete category over Ab (the

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: indeed, every small abelian category can be embedded in Ab. Properties The zero object o ...

) or over Mon (the

category of monoids In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms * ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'', * ''� ...

). Specifically, there are

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 every

abelian group In mathematics, 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 the order in which they are written. That is, the group operation is commu ...

''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 In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...

''X'' the integral monoid ring Z 'X''

Properties

Limits and colimits

The category Ring is both complete and cocomplete, 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 colimit In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered c ...

s, but does not preserve either

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...

s or

coequalizer In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. Definition A coequalizer is the ...

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 An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s Z is an

initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...

in Ring. *The

zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which fo ...

is a

terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

in Ring. *The product in Ring is given by the direct product of rings. This is just the

cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

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 In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, an ...

of groups. The coproduct of nonzero rings can be the zero ring; in particular, this happens whenever the factors have

relatively prime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...

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 In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...

). *The

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 kernel pair of ''f'' (this is just the pullback of ''f'' with itself) is a

congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the ...

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 morphisms (see below).

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

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: *

Isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

s in Ring are the

bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

ring homomorphisms. *

Monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphis ...

s in Ring are the

injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...

homomorphisms. Not every monomorphism is regular however. *Every surjective homomorphism is an

epimorphism In category theory, an epimorphism is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \implies g_1 = g_2. Epimorphisms are categorical analo ...

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 or 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

(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 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

in Ring is a

commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...

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

Subcategories

The category of rings has a number of important subcategories. These include the full subcategories of commutative rings,

integral domain In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...

principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...

s, and fields.

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, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

. 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 a

reflective subcategory In mathematics, a full 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'', or ''localization''. Dually, ''A'' ...

of Ring. The free commutative ring on a set of generators ''E'' is the

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...

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, the coproduct of two nonzero commutative rings can be zero. The

opposite category In category theory, a branch of mathematics, the opposite category or dual category C^ of a given Category (mathematics), category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal ...

of CRing is equivalent to the category of affine schemes. The equivalence is given by the

contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

Spec which sends a commutative ring to its

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

, 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 In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphis ...

. 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 In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...

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 a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

. Each such subcategory has an

: the

prime field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is wid ...

of characteristic ''p'' (which is Q if ''p'' = 0, otherwise the

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

F_''p'').

Related categories and functors

Category of groups

There is a natural functor from Ring to the

category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories The ...

, Grp, which sends each ring ''R'' to its

group of units In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the ele ...

''U''(''R'') and each ring homomorphism to the restriction to ''U''(''R''). This functor has a

which sends each group ''G'' to the integral group ring Z 'G'' 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 In mathematics, the projective line over a ring is an extension of the concept of projective line over a field (mathematics), field. Given a ring (mathematics), ring ''A'' (with 1), the projective line P1(''A'') over ''A'' consists of points iden ...

P(''R'').

''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 homomorphism In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...

s. 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 In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

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 In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...

''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 In mathematics, specifically category theory, 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, ...

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

serves as both an initial and terminal object in Rng (that is, it is a

zero object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

). It follows that Rng, like Grp but unlike Ring, has zero morphisms. These are just the rng homomorphisms that map everything to 0. Despite the existence of zero morphisms, Rng is still not a preadditive category. The pointwise sum of two rng homomorphisms is generally not a rng homomorphism. There is a fully faithful functor from the category of

s 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 is the ring of all integral polynomials over ''x'' with no constant term, while the free ring generated by is just the

Z 'x''

References

* * *{{cite book , first = Saunders , last = Mac Lane , year = 1998 , title =

Categories for the Working Mathematician ''Categories for the Working Mathematician'' (''CWM'') is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based ...

, series = Graduate Texts in Mathematics , volume=5 , edition = 2nd , publisher = Springer , isbn = 0-387-98403-8 Rings Ring theory