Definition: A ring R is a domain if it has no zero divisors. R is an integral domain if it is a commutative domain

Examples:

1. $\Z$ is an integral domain
2. If R is a division ring then it is a domain

(Because in a division ring, non-zero elements are units, but we saw that units cannot be zero-divisors!)

1. If R is a field then it is an integral domain

$$ \begin{matrix} Field & \implies & Integral\ Domain \\ \Downarrow & & \Downarrow \\ Division Ring & \implies & Domain \end{matrix} $$

Properties of Domains

1. Domains are “canellative” If $ab=ac,\ a\neq0 \implies b=c$

Proof:

Suppose that $ab=ac,\ a\neq 0$ in a domain. Then, $a(b-c)=ab-ac=0$. In a domain, if a product of two elements is 0 then one of them must be 0. Since $a\neq 0$, it follows that $b-c=0$. Namely, $b=c$

Remark: If your ring is not a domain, do not expect it to be cancellative

In $\Z_6$, $3\cdot 2 =3 \cdot 4$ but $3\neq 4$

Proof: Suppose R is a domain $S \subseteq R$ subring

Let $a,b \in S$ be non-zero, let us prove that $ab\neq 0$. But $a,b\in R$ are non-zero, and R is a domain, so $ab \neq 0$

If in addition R is commutative, then so is S

Remark: A subring of a field need not be a field Ex. Not a field → $\Z \sub R$ ← Field

Corollary: $\Z, \Z[{\sqrt2}], \Z[{i}], \cdots \subset \Complex$ therefore they are integral domains

Proposition: If R is a domain then $R[x]$ is also a domain

(And if R is an integral domain then so is $R[x]$