Integral Domains

Overview

An integral domain is a commutative ring with unity $1\ne 0$ and containing no divisors of 0.

Every finite integral domain is a field.

Subdomains

Let $⟨D,+,\cdot ⟩$ be an integral domain. Then $S$ is a subdomain of $D$ if:

• $S$ is a subring of $D$ and
• contains no divisors of $0$.

Set $\{n\cdot 1\mid n\in \mathbb{Z}\}$ forms a subdomain of $D$ under the induced addition and multiplication operations. It is contained by every subdomain of $D$.

Intersection

Let $\{S_i\mid i\in I\}$ be an indexed set of subdomains of integral domain $D$. Then their intersection $\bigcap _{i\in I}{S}_i$ is a subdomain of $D$.

Special Cases

Integers

The set of all integers is denoted $\mathbb{Z}$. The following notation is often used to denote particular subsets of $\mathbb{Z}$:

• ${\mathbb{Z}}_n=\{0,1,\dots ,n-1\}$ where $n\in \mathbb{N}$.
• ${\mathbb{Z}}^{+}=\{x\in \mathbb{Z}\mid x>0\}$
• $n\mathbb{Z}=\{nx\mid x\in \mathbb{Z}\}$ where $n\in \mathbb{N}$.

Set $\mathbb{Z}$, alongside the usual addition and multiplication, forms a ring $⟨\mathbb{Z},+,\cdot ⟩$.