Table of Contents
Integral Domain
Abbreviation: IntDom
Definition
An \emph{integral domain} is a commutative rings with identity $\mathbf{R}=\langle R,+,-,0,\cdot,1\rangle$ that
has no zero divisors: $\forall x,y\ (x\cdot y=0\Longrightarrow x=0\ \mbox{or}\ y=0)$
Morphisms
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 $h(-x)=-h(x)$.
Examples
Example 1: $\langle\mathbb{Z},+,-,0,\cdot,1\rangle$, the ring of integers with addition, subtraction, zero, and multiplication is an integral domain.
Basic results
Every finite integral domain is a fields.
Properties
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &1
f(5)= &1
f(6)= &0
\end{array}$
