Integral domain information

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In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.^[1]^[2] Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity.^[3]^[4] Noncommutative integral domains are sometimes admitted.^[5] This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings.

Some sources, notably Lang, use the term entire ring for integral domain.^[6]

Some specific kinds of integral domains are given with the following chain of class inclusions:

rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields

Algebraic structures
Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory
Ring-like Ring Rng Semiring Near-ring Commutative ring Domain Integral domain Field Division ring Lie ring Ring theory
Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map of lattices Lattice theory
Module-like Module Group with operators Vector space Linear algebra
Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra
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^ Bourbaki 1998, p. 116
^ Dummit & Foote 2004, p. 228
^ van der Waerden 1966, p. 36
^ Herstein 1964, pp. 88–90
^ McConnell & Robson
^ Lang 1993, pp. 91–92

Integral domain information

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Integral domain

Integrally closed domain

Multiple integral

Domain

Principal ideal domain

Unique factorization domain

Integral element

Integral transform

Improper integral

Euclidean domain

Dedekind domain

Laplace transform

Atomic domain

Integral

GCD domain

Irreducible polynomial

Fractional ideal

Volume integral

Functional integration

Field of fractions

Irreducible element

Square root

Transcendental extension

Lebesgue integration

Polynomial ring

Monic polynomial