# Definition:Integral Domain

*This page is about the concept of integral domain in ring theory. For other uses, see Definition:Domain.*

## Contents

## Definition

### Definition 1

An **integral domain** $\left({D, +, \circ}\right)$ 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 (from the Cancellation Law of Ring Product of Integral Domain) 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 $\left({D, *, \circ}\right)$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:

\((A0)\) | $:$ | Closure under addition | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle a * b \in D \) | ||||

\((A1)\) | $:$ | Associativity of addition | \(\displaystyle \forall a, b, c \in D:\) | \(\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right) \) | ||||

\((A2)\) | $:$ | Commutativity of addition | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle a * b = b * a \) | ||||

\((A3)\) | $:$ | Identity element for addition: the zero | \(\displaystyle \exists 0_D \in D: \forall a \in D:\) | \(\displaystyle a * 0_D = a = 0_D * a \) | ||||

\((A4)\) | $:$ | Inverse elements for addition: negative elements | \(\displaystyle \forall a \in D: \exists a' \in D:\) | \(\displaystyle a * a' = 0_D = a' * a \) | ||||

\((M0)\) | $:$ | Closure under product | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle a \circ b \in D \) | ||||

\((M1)\) | $:$ | Associativity of product | \(\displaystyle \forall a, b, c \in D:\) | \(\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right) \) | ||||

\((D)\) | $:$ | Product is distributive over addition | \(\displaystyle \forall a, b, c \in D:\) | \(\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right) \) | ||||

\(\displaystyle \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({b \circ c}\right) \) | ||||||||

\((C)\) | $:$ | Product is commutative | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle a \circ b = b \circ a \) | ||||

\((U)\) | $:$ | Identity element for product: the unity | \(\displaystyle \exists 1_D \in D: \forall a \in D:\) | \(\displaystyle a \circ 1_D = a = 1_D \circ a \) | ||||

\((ZD)\) | $:$ | No proper zero divisors | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle a \circ b = 0_D \iff a = 0 \lor b = 0 \) |

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 set theory, 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.