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 rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields
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
Famous quotes containing the words integral and/or domain:
“... no one who has not been an integral part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.”
—Angelina Grimké (1805–1879)
“Every sign is subject to the criteria of ideological evaluation.... The domain of ideology coincides with the domain of signs. They equate with one another. Wherever a sign is present, ideology is present, too. Everything ideological possesses semiotic value.”
—V.N. (Valintin Nikolaevic)