integral domain

An integral domain is a commutative ring with a unique zero divisor.

Formally,
An integral domain is a set $D$ equipped with two binary operations, addition and multiplication, such that for all elements $a,b,c\in D$,

$(D,+,\times )$ is a ring:

1. $(D,+)$ is closed: $a+b\in D$
2. $(D,+)$ is associative: $(a+b)+c=a+(b+c)$
3. $(D,+)$ has an identity element: $a+0=a=0+a$
4. $(D,+)$ has inverses: $a+(-a)=0=(-a)+a$
5. $(D,+)$ is commutative: $a+b=b+a$
6. $(D,\times )$ is closed: $ab\in D$
7. $(D,\times )$ is associative: $(ab)c=a(bc)$
8. $(D,\times )$ has an identity element: $a1=a=1a$
9. $(D,+,\times )$ is left-distributive: $a(b+c)=ab+ac$
10. $(D,+,\times )$ is right-distributive: $(a+b)c=ac+bc$

Additionally,

11. $(D,\times )$ is commutative: $ab=ba$
12. Unique zero divisor: if $ab=0$, then $a=0$ or $b=0$