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Integral domain

In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, ab = ac implies b = c. Integral domains are generalizations of the ring of integers and provide a setting that is useful for studying…

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Overview

Definition

Examples

Non-examples

Divisibility, prime elements, and irreducible elements

Properties

Field of fractions

Algebraic geometry

Characteristic and homomorphisms

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Map overview Semantic statistics

Integral domain

Nodes95
Edges94
Triples51
Avg. degree1.98
Density0.021053
Components1

How this topic connects Entity context

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Integral domain

Top relations

related to Examples · 11
Integral domain → Artinian, Conversely, Every, For, If, In, Integrality, Rings, The, UFD, Wedderburn's
related to Non-examples · 9
Integral domain → Consider, For, If, In, MN, The, Then, This, To
related to Properties · 9
Integral domain → An, Another, Hilbert's, If, Let, R/P, The, Then, This
is a · 5
Integral domain → field, integral domain.If U, nonzero commutative ring in which the product of any two nonzero elements is nonzero, nonzero commutative ring with no nonzero zero divisors.An integral domain is a commutative ring in which the zero ideal, ring that is isomorphic to a subring of a field
related to Algebraic geometry · 4
Integral domain → Integral, It, The, This
related to Definition · 4
Integral domain → An, Elements, Equivalently, Given
related to Characteristic and homomorphisms · 3
Integral domain → Frobenius, If, The
related to Divisibility, prime elements, and irreducible elements · 2
Integral domain → Given, In
related to Field of fractions · 2
Integral domain → It, The
see also · 2
Integral domain → Dedekind, Hasse

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Important terminology

integral ring domain displaystyle nonzero field domains mathbb prime product irreducible elements commutative element rings ideal example every integers zero

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Integral domainis anonzero commutative ring in which the product of any two nonzero elements is nonzero0.90text
Integral domainis anonzero commutative ring with no nonzero zero divisors.An integral domain is a commutative ring in which the zero ideal0.90text
Integral domainis aring that is isomorphic to a subring of a field0.90text
Integral domainis afield0.90text
Integral domainis aintegral domain.If U0.90text
Integral domainrelated to Algebraic geometryIntegral0.60section
Integral domainrelated to Algebraic geometryThe0.60section
Integral domainrelated to Algebraic geometryIt0.60section
Integral domainrelated to Algebraic geometryThis0.60section
Integral domainrelated to Characteristic and homomorphismsThe0.60section
Integral domainrelated to Characteristic and homomorphismsIf0.60section
Integral domainrelated to Characteristic and homomorphismsFrobenius0.60section

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