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

Other articles related to "integral domain, domain, domains":

Integral Domain - Characteristic and Homomorphisms
... The characteristic of every integral domain is either zero or a prime number ... If R is an integral domain with prime characteristic p, then f(x) = x p defines an injective ring homomorphism f R → R, the Frobenius endomorphism ...
Gauss's Lemma (polynomial) - Proofs of The Primitivity Statement
... proof can be given using the statement from abstract algebra that a polynomial ring over an integral domain is again an integral domain ... pR generated by p is a prime ideal, so R/pR is an integral domain, and (R/pR) is therefore an integral domain as well ... reduction modulo p kills the uninteresting terms what is left is a proof that polynomials over an integral domain cannot be zero divisors by ...
Algebraic Number Field - Algebraicity and Ring of Integers
... Therefore, the ring of integers of F is an integral domain ... The field F is the field of fractions of the integral domain OF ... of algebraic integers have three distinctive properties firstly, OF is an integral domain that is integrally closed in its field of fractions F ...
Atomic Domain
... In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain, every non-zero non-unit of which can be written (in at least one way) as a (finite) product of ... Atomic domains different from unique factorization domains in that this decomposition of an element into irreducibles need not be unique stated differently, an ... Important examples of atomic domains include the class of all unique factorization domains, and all Noetherian domains ...
Absolute Value (algebra) - Completions
... Given an integral domain D with an absolute value, we can define the Cauchy sequences of elements of D with respect to the absolute value by requiring that for every r > 0 ... ideal in the ring of Cauchy sequences, and the quotient ring is therefore an integral domain ... The domain D is embedded in this quotient ring, called the completion of D with respect to the absolute value

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