Abbreviation: IntDom

An \emph{integral domain} is a commutative rings with identity $\mathbf{R}=\langle R,+,-,0,\cdot,1\rangle$ that

has no zero divisors: $\forall x,y\ (x\cdot y=0\Longrightarrow x=0\ \mbox{or}\ y=0)$

Let $\mathbf{R}$ and $\mathbf{S}$ be integral domains. A morphism from $\mathbf{R}$ to $\mathbf{S}$ is a function $h:R\rightarrow S$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

Remark: It follows that $h(0)=0$ and $h(-x)=-h(x)$.

Example 1: $\langle\mathbb{Z},+,-,0,\cdot,1\rangle$, the ring of integers with addition, subtraction, zero, and multiplication is an integral domain.

Every finite integral domain is a fields.

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= &1 f(5)= &1 f(6)= &0 \end{array}$

Unique factorization domains

Commutative rings with identity