Integral domain

This article gives a basic definition in the following area: ring theory
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Definition

In words

An integral domain is a ring in which any product of non-zero elements is non-zero.

In symbols

An integral domain is a ring $R$ such that if $a,b\in R$ with $ab=0$, then $a=0$ or $b=0$.

Relation to other properties

Stronger properties

• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field

Weaker properties

• Ring

Examples and non-examples

Examples

• The ring of integers is an integral domain.

Non-examples

• The rings of integers modulo composite numbers are not integral domains. For example, the ring of integers modulo four is not an integral domain, as $2\cdot 2=0$.