# Integral domain

This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
This page needs an update to the modern style of the site, it was also created largely as a stub page

## Definition

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

• A commutative ring, that is: $\forall x,y\in D[xy=yx]$
• 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:
• $\exists c\in R[c\ne e_+\wedge ac=e_+]$ or if (by writing [ilmath]e_+[/ilmath] as [ilmath]0[/ilmath] we can say: $\exists c\in R[c\ne 0\wedge ac=0]$)
• $\exists d\in R[d\ne e_+\wedge da=e_+]$ (by writing [ilmath]e_+[/ilmath] as [ilmath]0[/ilmath] we can say: $\exists d\in R[d\ne 0\wedge da=0]$)
• We can write this as: $\exists c\in R[c\ne 0\wedge(ac=0\vee ca=0)]$

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 a ring [ilmath]D[/ilmath] is an integral domain if:

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

## 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]===e_+[/ilmath].

This means both [ilmath][/ilmath] and [ilmath][/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