Domain

definitionring-theory

Definition

A ring $(R,+,\cdot )$ is called a domain (integral domain) if $(0)⊴R$ is a prime ideal.

Equivalently, $xy=0⟹x=0$ or $y=0$.

Equivalently, the only zero-divisor in $R$ is $0_R=0$.

Examples

• $(\mathbb{Z},+,\cdot ),(\mathbb{Q},+,\cdot ),(\mathbb{R},+,\cdot )$ are domains.
• For $n\ge 2$, $({\mathbb{Z}}_n,+,\cdot )$ is a domain if and only if $n$ is a prime integer.