Definition:Integral Domain
This page is about integral domain in the context of ring theory. For other uses, see domain.
Definition
Definition $1$
Let $\struct {D, +, \circ}$ be an algebraic structure.
$\struct {D, +, \circ}$ is an integral domain if and only if it fulfils the conditions of the integral domain axioms, as follows:
An integral domain is an algebraic structure $\struct {D, +, \circ}$, on which are defined two binary operations $\circ$ and $+$, which satisfy the following conditions:
| \((\text A 0)\) | $:$ | Closure under addition | \(\ds \forall a, b \in D:\) | \(\ds a + b \in D \) | |||||||
| \((\text A 1)\) | $:$ | Associativity of addition | \(\ds \forall a, b, c \in D:\) | \(\ds \paren {a + b} + c = a + \paren {b + c} \) | |||||||
| \((\text A 2)\) | $:$ | Commutativity of addition | \(\ds \forall a, b \in D:\) | \(\ds a + b = b + a \) | |||||||
| \((\text A 3)\) | $:$ | Identity element for addition: the zero | \(\ds \exists 0_D \in D: \forall a \in D:\) | \(\ds a + 0_D = a = 0_D + a \) | |||||||
| \((\text A 4)\) | $:$ | Inverse elements for addition: negative elements | \(\ds \forall a \in D: \exists a' \in D:\) | \(\ds a + a' = 0_D = a' + a \) | |||||||
| \((\text M 0)\) | $:$ | Closure under product | \(\ds \forall a, b \in D:\) | \(\ds a \circ b \in D \) | |||||||
| \((\text M 1)\) | $:$ | Associativity of product | \(\ds \forall a, b, c \in D:\) | \(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | |||||||
| \((\text M 2)\) | $:$ | Commutativity of product | \(\ds \forall a, b \in D:\) | \(\ds a \circ b = b \circ a \) | |||||||
| \((\text M 3)\) | $:$ | Non-zero identity element for product: the unity | \(\ds \exists 1_D \in D: 1_D \ne 0_D: \forall a \in D:\) | \(\ds a \circ 1_D = a = 1_D \circ a \) | |||||||
| \((\text D)\) | $:$ | Product is distributive over addition | \(\ds \forall a, b, c \in D:\) | \(\ds a \circ \paren {b + c} = \paren {a \circ b} + \paren {a \circ c} \) | |||||||
| \(\ds \paren {a + b} \circ c = \paren {a \circ c} + \paren {b \circ c} \) | |||||||||||
| \((\text C)\) | $:$ | $\struct {D, +, \circ}$ has no (proper) zero divisors | \(\ds \forall a, b \in D:\) | \(\ds a \circ b = 0_D \implies a = 0_D \text{ or } b = 0_D \) |
These criteria are called the integral domain axioms.
Definition $2$
An integral domain $\struct {D, +, \circ}$ is:
- a commutative ring with unity
- in which there are no (proper) zero divisors, that is:
- $\forall x, y \in D: x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D$
that is, in which all non-zero elements are cancellable.
Definition $3$
An integral domain $\struct {D, +, \circ}$ is a commutative ring with unity all of whose non-zero elements are cancellable.
Also defined as
Some authors do not specify that an integral domain be commutative.
Some sources do not insist that an integral domain needs to be a non-null ring, and hence (either intentionally or by accident) allow a null ring to be classified as an integral domain.
Also known as
Some authors refer to an integral domain as simply a domain.
However, this conflicts with the concept of domain in the context of mappings and relations.
Therefore, it is always best to refer to an integral domain, as to avoid possible confusion.
Also see
- Results about integral domains can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): domain: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): domain: 2.
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): domain (algebra)