# Integral domain

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## Contents

## Definition

Given a ring [ilmath](D,+,\times)[/ilmath], it is called an *integral domain*^{[1]} if it is:

- A commutative ring, that is: [math]\forall x,y\in D[xy=yx][/math]
- Contains no
*non-zero divisors of zero*- An element [ilmath]a[/ilmath] of a ring [ilmath]R[/ilmath] is said to be a divisor of zero in [ilmath]R[/ilmath] if:
- [math]\exists c\in R[c\ne e_+\wedge ac=e_+][/math] or if (by writing [ilmath]e_+[/ilmath] as [ilmath]0[/ilmath] we can say: [math]\exists c\in R[c\ne 0\wedge ac=0][/math])
- [math]\exists d\in R[d\ne e_+\wedge da=e_+][/math] (by writing [ilmath]e_+[/ilmath] as [ilmath]0[/ilmath] we can say: [math]\exists d\in R[d\ne 0\wedge da=0][/math])
- We can write this as: [math]\exists c\in R[c\ne 0\wedge(ac=0\vee ca=0)][/math]

- An element [ilmath]a[/ilmath] of a ring [ilmath]R[/ilmath] is said to be a divisor of zero in [ilmath]R[/ilmath] if:

As the *integral domain* is commutative we don't need both [ilmath]ac[/ilmath] and [ilmath]ca[/ilmath].

### Shorter definition

We can restate this as^{[2]} a ring [ilmath]D[/ilmath] is an integral domain if:

- [math]\forall x,y\in D[xy=yx][/math]
- [math]\forall a,b\in D[(a\ne 0,b\ne 0)\implies(ab\ne 0)][/math]

## Example of a ring that isn't an integral domain

Consider the ring [ilmath]\mathbb{Z}/6\mathbb{Z} [/ilmath], the ring of integers modulo 6, notice that [ilmath][2][3]=[6]=[0]=e_+[/ilmath].

This means both [ilmath][2][/ilmath] and [ilmath][3][/ilmath] are *non-zero* divisors of zero.

## Examples of rings that are integral domains

- The integers
- [ilmath]\mathbb{Z}/p\mathbb{Z} [/ilmath] where [ilmath]p[/ilmath] is prime

## See next

## See also