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In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously

Contents

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 preserving this structure. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function There is a natural forgetful functor

U : Ring → Set

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). In Mathematics, in the area of Category theory, a forgetful functor is a type of Functor. In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are This functor has a left adjoint

F : Set → Ring

which assigns to each set X the free ring generated by X. In Mathematics, especially in the area of Abstract algebra known as Ring theory, a free algebra is the noncommutative analogue of a Polynomial ring

One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of monoids). In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype In Mathematics, the category of magmas (see category, magma for definitions denoted by Mag, has as objects sets with a Binary operation Specifically, there are faithful functors

A : Ring → Ab
M : Ring → Mon

which "forget" multiplication and addition, respectively. In Category theory, a faithful functor (resp a full functor) is a Functor which is Injective (resp Both of these functors have left adjoints. 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). An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra The left adjoint of M is the functor which assigns to every monoid X the integral monoid ring Z[M]. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Abstract algebra, a monoid ring is a new ring constructed from some other ring and a Monoid.

Properties

Limits and colimits

The category Ring is both complete and cocomplete, meaning that all small limits and colimits exist in Ring. In Mathematics, a complete category is a category in which all small limits exist In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts Like many other algebraic categories, the forgetful functor U : Ring → Set creates (and preserves) limits and filtered colimits, but does not preserve either coproducts or coequalizers. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts In Category theory, filtered categories generalize the notion of Directed set. In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological In Mathematics, a coequalizer (or coequaliser) is a generalization of a quotient by an Equivalence relation to objects in an arbitrary category The forgetful functors to Ab and Mon also create and preserve limits.

Examples of limits and colimits in Ring include:

Morphisms

Unlike many categories studied in mathematics, there do not always exist morphisms between pairs of objects in Ring. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In Mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects the precise In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers 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 trivial ring is a ring defined on a Singleton set, { r } The ring operations (× and + are trivial r \times r = r A necessary condition for there to be morphisms from R to S is that the characteristic of S divide that of R. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's

Note that even though some of the hom-sets are empty, the category Ring is still connected since it has an initial object. In Category theory, a branch of Mathematics, a connected category is a category in which for every two objects X and Y there is a

Some special classes of morphisms in Ring include:

Other properties

Subcategories

The category of rings has a number of important subcategories. In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in These include the full subcategories of commutative rings, integral domains, principal ideal domains, and fields. In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

Category of commutative rings

The category of commutative rings, denoted CRing, is the full subcategory of Ring whose objects are all commutative rings. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property This category is one of the central objects of study in the subject of commutative algebra. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings

Any ring can be made commutative by taking the quotient by the ideal generated by all elements of the form (xy − yx). In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. This defines a functor Ring → CRing which is left adjoint to the inclusion functor, so that CRing is a reflective subcategory of Ring. In Mathematics, a Subcategory A of a category B is said to be reflective in B when the Inclusion functor from The free commutative ring on a set of generators E is the polynomial ring Z[E] whose variables are taken from E. 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 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 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 the tensor product of rings. In Mathematics, the Tensor product of two ''R''-algebras is also an R -algebra in a natural way Again, its quite possible for the coproduct of two nontrivial commutative rings to be trivial.

The opposite category of CRing is equivalent to the category of affine schemes. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of The equivalence is given by the contravariant functor Spec which sends a commutative ring to its spectrum, an affine scheme. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory.

Category of fields

The category of fields, denoted Field, is the full subcategory of CRing whose objects are fields. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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 Mathematics, a complete category is a category in which all small limits exist 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 simply an Injective Homomorphism. This follows from the fact that the only ideals in a field F are the zero ideal and F itself. The number 0 is an important concept in Mathematics. Zero module In Mathematics, the zero module is the module consisting of only One can then view morphisms in Field as field extensions. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory.

The category of fields is not connected. In Category theory, a branch of Mathematics, a connected category is a category in which for every two objects X and Y there is a There are no morphisms between fields of different characteristic. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's The connected components of Field are the full subcategories of characteristic p, where p = 0 or is a prime number. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Each such subcategory has an initial object: the prime field of characteristic p (which is Q if p = 0, otherwise the finite field Fp). In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements

Related categories and functors

Category of groups

There is a natural functor from Ring to the category of groups, Grp, which sends each ring R to its group of units U(R) and each ring homomorphism to the restriction to U(R). In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i This functor has a left adjoint which sends each group G to the integral group ring Z[G]. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively

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. 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

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. In Category theory, two categories C and D are isomorphic if there exist Functors F: C &rarr D and G 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⊗ZA, thought of as an R-algebra by setting r·(s⊗a) = rs⊗a. In Mathematics, the Tensor product of two ''R''-algebras is also an R -algebra in a natural way

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. In Abstract algebra, a rng (also called a pseudo-ring or non-unital ring) is an Algebraic structure satisfying the same properties as a The category of all rngs will be denoted by Rng.

The category of rings, Ring, is a nonfull subcategory of Rng. In Mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in 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. In Mathematics, a Subcategory A of a category B is said to be reflective in B when the Inclusion functor from

The trivial ring serves as both a initial and terminal object in Rng (that is, it is a zero object). A trivial ring is a ring defined on a Singleton set, { r } The ring operations (× and + are trivial r \times r = r It follows that Rng, like Grp but unlike Ring, has zero morphisms. In Category theory, a zero morphism is a special kind of "trivial" Morphism. 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 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 of two rngs is given by a direct sum construction analogous to that of abelian groups. In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

Free constructions are less natural in Rng then they are in Ring. In Mathematics, the idea of a free object is one of the basic concepts of Abstract algebra. 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 Z[x]. 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

References

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