# Definition:Integral Domain

*This page is about Integral Domain in the context of Ring Theory. For other uses, see Domain.*

## Definition

### Definition 1

An **integral domain** $\struct {D, +, \circ}$ is:

- a commutative ring which is non-null
- with a 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 2

An **integral domain** $\left({D, +, \circ}\right)$ is a commutative ring such that $\left({D^*, \circ}\right)$ is a monoid, all of whose elements are cancellable.

In this context, $D^*$ denotes the ring $D$ without zero: $D \setminus \left\{{0_D}\right\}$.

### Integral Domain Axioms

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 | \(\ds \forall a, b \in D:\) | \(\ds a \circ b = b \circ a \) | |||||

\((\text M 3)\) | $:$ | Identity element for product: the unity | \(\ds \exists 1_D \in D: \forall a \in D:\) | \(\ds a \circ 1_D = a = 1_D \circ a \) | |||||

\((\text D)\) | $:$ | 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 | \(\ds \forall a, b \in D:\) | \(\ds x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D \) |

These criteria are called the **integral domain axioms**.

## Also known as

Some authors refer to this concept 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 defined as

Some authors do not require that an **integral domain** be commutative.

## Also see

- Results about
**integral domains**can be found**here**.