Integral Domain

In abstract algebra, an integral domain is a commutative ring that has no zero divisors, and which is not the trivial ring {0}. It is usually assumed that commutative rings and integral domains have a multiplicative identity even though this is not always included in the definition of a ring. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain with identity.

The above is how "integral domain" is almost universally defined, but there is some variation. In particular, noncommutative integral domains are sometimes admitted. However, this article follows the much more usual convention of reserving the term integral domain for the commutative case and using domain for the noncommutative case; curiously, the adjective "integral" implies "commutative" in this context. Some sources, notably Lang, use the term entire ring for integral domain.

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

Commutative ringsintegral domainsintegrally closed domainsunique factorization domainsprincipal ideal domainsEuclidean domainsfields

The absence of zero divisors means that in an integral domain the cancellation property holds for multiplication by any nonzero element a: an equality ab = ac implies b = c.

Algebraic structures
Group-like structures Semigroup and Monoid
Quasigroup and Loop
Abelian group
Ring-like structures Semiring
Near-ring
Ring
Commutative ring
Integral domain
Field
Lattice-like structures Semilattice
Lattice
Map of lattices
Module-like structures Group with operators
Module
Vector space
Algebra-like structures Algebra
Associative algebra
Non-associative algebra
Graded algebra
Bialgebra

Read more about Integral Domain:  Definitions, Examples, Divisibility, Prime Elements, and Irreducible Elements, Properties, Field of Fractions, Algebraic Geometry, Characteristic and Homomorphisms

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