Ring (mathematics)

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Basic examples

Algebra over a commutative ring R-module Formal power series Continuous Pointwise

26 related topics

Basic concepts

Nilpotent element Nilpotent matrix Idempotent Projection Unit

30 related topics

Constructions

Product Direct product Chinese remainder theorem Central idempotents Projective limit

48 related topics

Special kinds of rings

Domain Prime elements Number field Structure theorem for finitely generated modules over a principal ideal domain Cyclic modules

27 related topics

Topics to explore

Browse the full topic structure. Each item opens a new analysis centered on that subject.

Overview

Definition

Set Set (mathematics)
Associative Associativity
Commutative Commutativity
Additive identity
Additive inverse
Monoid
Multiplicative identity
Distributive Distributive law
Even integers Even integer
Commutative Commutative property
Nonzero Zero ring
Multiplicative inverse
Algebra Algebra over a field

Illustration

Modules

Vector space
Scalar multiplication
Operation Operation (mathematics)
Axioms Axiom
Noncommutative Noncommutative ring
Dimension of a vector space Dimension (vector space)
Basis Basis (linear algebra)
Center Center of a ring

Constructions

Product Cartesian product
Direct product Direct product of rings
Chinese remainder theorem
Central idempotents Central idempotent
Projective limit
Restricted product
Adele ring
Rational functions Rational function
Substitution Substitution (algebra)
Symmetric algebra
Polynomial function
Hilbert's Nullstellensatz
Gröbner basis
Formal power series ring
Local ring
Complete Complete ring
Composition of functions
Schur's lemma
Semisimple ring
Morita equivalent
Category Category (mathematics)
Filtered colimit
Algebraic closure
Finite fields Finite field
Function field of an algebraic variety
Stalk Stalk (mathematics)
Generic point
Finitely generated subrings Finitely generated ring
Filtered limit
Localization Localization of a ring

Special kinds of rings

Domain Domain (ring theory)
Prime elements Prime element
Number field
Structure theorem for finitely generated modules over a principal ideal domain
Cyclic modules Cyclic module
Jordan matrix
Jordan canonical form
Regular local ring
Quaternions Quaternion
Wedderburn's little theorem
Cartan–Brauer–Hua theorem
Cyclic algebra
L. E. Dickson
Quaternion algebra
Semisimple module
Skolem–Noether theorem
Brauer group
Quasi-algebraically closed field
Tsen's theorem
Theorem of Frobenius Frobenius theorem (real division algebras)
Local field
Invariant map Hasse invariant of an algebra
Azumaya algebras Azumaya algebra
Valuation Valuation (algebra)
Valuation ring
Formal Laurent series
Well ordered

Rings with extra structure

Topological ring
Topology Topological space
Continuous Continuous function (topology)
Product topology
Euclidean topology
Zariski topology
Λ-ring
Exterior powers Exterior power
Binomial coefficients Binomial coefficient
Riemann–Roch theorem
Totally ordered ring
Total ordering

Some examples of the ubiquity of rings

Mathematical objects Mathematical object
Graded ring
Homology groups Homology group
Spheres Sphere
Tori Torus
Point-set topology
Cohomology groups Cohomology group
Universal coefficient theorem
Natural product Cup product
Multilinear form
Characteristic classes Characteristic class
Fiber bundles Fiber bundle
Algebraic variety
Schubert calculus
Burnside ring
Act Group action (mathematics)
Free abelian group
Representation ring
Hopf algebra
Character theory
Grothendieck group
Birational geometry
Simplicial complex
Stanley–Reisner ring
Algebraic combinatorics
Simplicial polytopes Simplicial polytope

Category-theoretic description

Monoid Monoid (category theory)
Category of abelian groups
Monoidal category
Tensor product of ⁠ Z {\displaystyle \mathbb {Z} } ⁠-modules Tensor product of abelian groups
Endomorphisms Endomorphism
Set of operators Group with operators
Preadditive category
Additive functors Additive functor
Morphisms Morphism

Generalization

Other ring-like objects

Products Product (category theory)
Terminal object
Scheme Scheme (mathematics)
Witt vectors Witt vector
Algebraic topology
Ring spectrum
Spectrum Spectrum (topology)
Sphere spectrum
Monoid object
Symmetric spectra Symmetric spectrum

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

Map overview Semantic statistics

Nodes307

Edges306

Triples13

Avg. degree1.99

Density0.006515

Components1

How this topic connects Entity context

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Important terminology Word statistics

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

ring displaystyle commutative rings set field multiplication group algebra called ideal addition left element right homomorphism example integers mathbb one

Entity relationships Subject–Predicate–Object triples

Subject	Predicate	Object	Confidence	Src
integers or complex numbers	instance of	Ring elements may be numbers	0.80	text
but they may also be non-numerical objects such as polynomials	instance of	Ring elements may be numbers	0.80	text
square matrices	instance of	Ring elements may be numbers	0.80	text
functions	instance of	Ring elements may be numbers	0.80	text
and power series.More formally	instance of	Ring elements may be numbers	0.80	text
a ring is a set that is endowed with two binary operations	instance of	Ring elements may be numbers	0.80	text
geometry	instance of	They later proved useful in other branches of mathematics	0.80	text
analysis.Rings appear in the following chain of class inclusions	instance of	They later proved useful in other branches of mathematics	0.80	text
Artin	instance of	especially in advanced books by notable authors	0.80	text
Bourbaki	instance of	especially in advanced books by notable authors	0.80	text
Eisenbud	instance of	especially in advanced books by notable authors	0.80	text
and Lang	instance of	especially in advanced books by notable authors	0.80	text

Related concept clusters Concept neighborhoods

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Connections between topic areas Semantic bridges

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Basic examples

Basic concepts

Constructions

Special kinds of rings

Key facts & relationships

Topics to explore

Overview

Definition

Illustration

History

Basic examples

Basic concepts

Modules

Constructions

Special kinds of rings

Rings with extra structure

Some examples of the ubiquity of rings

Category-theoretic description

Generalization

Other ring-like objects

Advanced semantic analysis

Map overview Semantic statistics